two heads are better than none or, me and the fibonaccisskennedy/kstwoheads.pdf · the problem the...
TRANSCRIPT
![Page 1: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/1.jpg)
Two Heads Are Better Than NoneOr, Me and the Fibonaccis
Steve Kennedy(with help from Matt Stafford)
Carleton College andMAA Books
Problems are the lifeblood of mathematics.
— David Hilbert
Indiana MAA Spring Section Meeting 2014
![Page 2: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/2.jpg)
The Problem
The Game:
Flip a fair coin repeatedly until two consecutiveheads appear, stop.
The Problem:
I What is the probability that the game will ever end?(Intuitively, this is big, but consider HTHTHTHT. . ., orTTTTTT. . ..)
![Page 3: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/3.jpg)
The Problem
The Game:
Flip a fair coin repeatedly until two consecutiveheads appear, stop.
The Problem:
I What is the probability that the game will ever end?(Intuitively, this is big, but consider HTHTHTHT. . ., orTTTTTT. . ..)
![Page 4: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/4.jpg)
The Problem
The Game:
Flip a fair coin repeatedly until two consecutiveheads appear, stop.
The Problem(s):
I What is the probability that the game will ever end?(Intuitively, this is big, but consider HTHTHTHT..., orTTTTTT...)
I What is the probability that the game ends after exactly nflips?
![Page 5: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/5.jpg)
Counting Arguments
n = 2HH THHT TT
Each event equally likely, so P2 = 1/4.
![Page 6: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/6.jpg)
Counting Arguments
n = 2HH THHT TT
Each event is equally likely, so P2 = 1/4.
n=3
HHH THHHHT THTHTH TTHHTT TTT
Each event is equally likely, so P3 = 1/8.
![Page 7: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/7.jpg)
Counting Arguments: Wait just one minute!
n = 2HH THHT TT
Each event is equally likely, so P2 = 1/4.
n=3
HHH THHHHT THTHTH TTHHTT TTT
Each event is equally likely, so P3 = 1/8.
![Page 8: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/8.jpg)
Non–Counting Arguments
Yes, neither HHH nor HHT can actually happen. So, theprobability of seeing THH is really 1/6. Right?
![Page 9: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/9.jpg)
Non–Counting Arguments
Neither HHH nor HHT can actually happen. So, the probability ofseeing THH is really 1/6?
Well, not exactly. The probability of seeing THH is 1/6 times theprobability that the game didn’t end in exactly two flips.
![Page 10: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/10.jpg)
Non–Counting Arguments
Neither HHH nor HHT can actually happen. So, the probability ofseeing THH is really 1/6?
Well, not exactly. The probability of seeing THH is 1/6 times theprobability that the game didn’t end in exactly two flips.
So, actually, the probability of seeing THH is:
1
6× 6
8.
![Page 11: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/11.jpg)
Non–Counting Arguments
Yes, neither HHH nor HHT can actually happen. So, theprobability of seeing THH is really 1/6.
Well, not exactly. The probability of seeing THH is 1/6 times theprobability that the game didn’t end in exactly two flips.
So, actually, the probability of seeing THH is:
1
6× 6
8=
1
8.
This had to work out.
![Page 12: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/12.jpg)
Non–Counting Arguments
Yes, neither HHH nor HHT can actually happen. So, theprobability of seeing THH is really 1/6.
Well, not exactly. The probability of seeing THH is 1/6 times theprobability that the game didn’t end in exactly two flips.
So, actually, the probability of seeing THH is:
1
6× 6
8=
1
8.
This had to work out—because the coin doesn’t know the rules ofthe game.
![Page 13: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/13.jpg)
Counting Arguments
n = 2HH THHT TT
Each event is equally likely, so P2 = 1/4.
n=3
HHH THHHHT THTHTH TTHHTT TTT
Each event is equally likely, so P3 = 1/8.
n=4
HHHH HTHH THHH TTHHHHHT HTHT THHT TTHTHHTH HTTH THTH TTTHHHTT HTTT THTT TTTT
Each event is equally likely, so P4 = 2/16.
![Page 14: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/14.jpg)
Counting Arguments
n=5
HHHHH HTHHH THHHH TTHHHHHHHT HTHHT THHHT TTHHTHHHTH HTHTH THHTH TTHTHHHHTT HTHTT THHTT TTHTTHHTHH HTTHH THTHH TTTHHHHTHT HTTHT THTHT TTTHTHHTTH HTTTH THTTH TTTTHHHTTT HTTTT THTTT TTTTT
So, P5 = 3/32.
n=6 HTHTHH, HTTTHH, THTTHH, TTHTHH, TTTTHH
So, P6 = 5/64.
n=7 Homework
![Page 15: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/15.jpg)
Counting Arguments
n=5
HHHHH HTHHH THHHH TTHHHHHHHT HTHHT THHHT TTHHTHHHTH HTHTH THHTH TTHTHHHHTT HTHTT THHTT TTHTTHHTHH HTTHH THTHH TTTHHHHTHT HTTHT THTHT TTTHTHHTTH HTTTH THTTH TTTTHHHTTT HTTTT THTTT TTTTT
So, P5 = 3/32.
n=6 HTHTHH, HTTTHH, THTTHH, TTHTHH, TTTTHH
So, P6 = 5/64.
n=7 Homework
![Page 16: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/16.jpg)
Counting Arguments
n=5
HHHHH HTHHH THHHH TTHHHHHHHT HTHHT THHHT TTHHTHHHTH HTHTH THHTH TTHTHHHHTT HTHTT THHTT TTHTTHHTHH HTTHH THTHH TTTHHHHTHT HTTHT THTHT TTTHTHHTTH HTTTH THTTH TTTTHHHTTT HTTTT THTTT TTTTT
So, P5 = 3/32.
n=6 HTHTHH, HTTTHH, THTTHH, TTHTHH, TTTTHH
So, P6 = 5/64.
n=7
Homework
![Page 17: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/17.jpg)
Counting Arguments
n=5
HHHHH HTHHH THHHH TTHHHHHHHT HTHHT THHHT TTHHTHHHTH HTHTH THHTH TTHTHHHHTT HTHTT THHTT TTHTTHHTHH HTTHH THTHH TTTHHHHTHT HTTHT THTHT TTTHTHHTTH HTTTH THTTH TTTTHHHTTT HTTTT THTTT TTTTT
So, P5 = 3/32.
n=6 HTHTHH, HTTTHH, THTTHH, TTHTHH, TTTTHH
So, P6 = 5/64.
n=7 Homework
![Page 18: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/18.jpg)
Search for Pattern
14,
18,
216,
332,
564, . . .
Pn = xn2n , What’s xn?
xn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
![Page 19: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/19.jpg)
Search for Pattern
14,
18,
216,
332,
564, . . .
Pn = xn2n , What’s xn?
xn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
![Page 20: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/20.jpg)
Search for Pattern
14,
18,
216,
332,
564, . . .
Pn = xn2n , What’s xn?
xn = 1, 1, 2, 3, 5, 8, 13, 21, . . .
![Page 21: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/21.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 22: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/22.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 23: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/23.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 24: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/24.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 25: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/25.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 26: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/26.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
The Rules for Rabbit Reproduction
1. Gestation period is one month.
2. Rabbits born in male/female pairs.
3. Rabbits reach sexual maturity in one month.
4. Rabbits are always pregnant.(And never die.)
Start with one newborn pair, how many pairs will you have nmonths later?
![Page 27: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/27.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0Immature Pairs 1
Total Pairs 1
![Page 28: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/28.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0 1Immature Pairs 1
Total Pairs 1
![Page 29: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/29.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0 1Immature Pairs 1 0
Total Pairs 1 1
![Page 30: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/30.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0 1 1Immature Pairs 1 0 1
Total Pairs 1 1 2
![Page 31: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/31.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0 1 1 2Immature Pairs 1 0 1 1
Total Pairs 1 1 2 3
![Page 32: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/32.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202
Month: 0 1 2 3 4 5Mature Pairs 0 1 1 2 3 5Immature Pairs 1 0 1 1 2 3
Total Pairs 1 1 2 3 5 8
![Page 33: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/33.jpg)
An Unnecessary(?) Aside
The Fibonacci Numbers — 1202Month: 0 1 2 3 4 5Mature Pairs 0 1 1 2 3 5Immature Pairs 1 0 1 1 2 3
Total Pairs 1 1 2 3 5 8
Thus, Fn+1 = Fn + Fn−1.
![Page 34: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/34.jpg)
An Answer
TheoremThe sequence xn is the Fibonacci sequence.
Example
n=6T HTT HTT
![Page 35: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/35.jpg)
An Answer
TheoremThe sequence xn is the Fibonacci sequence.
Example
n=6THTTHH HTHTHHTTHTHH HTTTHHTTTTHH
So, x6 ≤ x5 + x4.
![Page 36: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/36.jpg)
An Answer
TheoremThe sequence xn is the Fibonacci sequence.
Example
n=6THTTHH HTHTHHTTHTHH HTTTHHTTTTHH
So, x6 ≤ x5 + x4.
Conversely*HTTHH **HTHH*THTHH **TTHH*TTTHH
![Page 37: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/37.jpg)
An Answer
TheoremThe sequence xn is the Fibonacci sequence.
Example
n=6THTTHH HTHTHHTTHTHH HTTTHHTTTTHH
So, x6 ≤ x5 + x4.
ConverselyTHTTHH HTHTHHTTHTHH HTTTHHTTTTHH
So, x6 ≥ x5 + x4.
![Page 38: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/38.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 39: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/39.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 40: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/40.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 41: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/41.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 42: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/42.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 43: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/43.jpg)
More Questions
I What is the probability that the game ends after two or threeflips?
1
4+
1
8=
3
8
I What is the probability that the game ends in four or fewerflips?
1
4+
1
8+
2
16=
1
2
I What is the probability that the game ends in twenty or fewerflips?
20∑n=2
Pn ≈ .983
![Page 44: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/44.jpg)
Original Question
What is the probability that the game ever ends?
limn→∞
n∑k=2
Pn =∞∑k=0
Fk
2k+2
Does this series converge?
![Page 45: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/45.jpg)
Original Question
What is the probability that the game ever ends?
limn→∞
n∑k=2
Pn =∞∑k=0
Fk
2k+2
Does this series converge?
![Page 46: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/46.jpg)
Original Question
What is the probability that the game ever ends?
limn→∞
n∑k=2
Pn =∞∑k=0
Fk
2k+2
Does this series converge?
![Page 47: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/47.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
![Page 48: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/48.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
2
1
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
![Page 49: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/49.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
8
1
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
![Page 50: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/50.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
16
1
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
![Page 51: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/51.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
32
1
128+ . . . =
1
64. . .
![Page 52: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/52.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
![Page 53: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/53.jpg)
Convergence
1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . . = S
1
4+
1
8+
1
16+
1
32+
1
64+
1
128+ . . . =
1
21
16+
1
32+
1
64+
1
128+ . . . =
1
81
32+
1
64+
1
128+ . . . =
1
161
64+
1
128+ . . . =
1
321
64+
1
128+ . . . =
1
321
128+ . . . =
1
64. . .
So, S = 12 + 1
8 + 116 + 2
32 + 364 + . . ..
![Page 54: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/54.jpg)
The Sum
Recall,
S =1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . .
We just decided,
S =1
2+
1
8+
1
16+
2
32+
3
64+
5
128+ . . .
So,
S =1
2+
1
2
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
S =1
2+
1
2S
S = 1
![Page 55: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/55.jpg)
The Sum
Recall,
S =1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . .
We just decided,
S =1
2+
1
8+
1
16+
2
32+
3
64+
5
128+ . . .
So,
S =1
2+
1
2
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
S =1
2+
1
2S
S = 1
![Page 56: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/56.jpg)
The Sum
Recall,
S =1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . .
We just decided,
S =1
2+
1
8+
1
16+
2
32+
3
64+
5
128+ . . .
So,
S =1
2+
1
2
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
S =1
2+
1
2S
S = 1
![Page 57: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/57.jpg)
The Sum
Recall,
S =1
4+
1
8+
2
16+
3
32+
5
64+
8
128+ . . .
We just decided,
S =1
2+
1
8+
1
16+
2
32+
3
64+
5
128+ . . .
So,
S =1
2+
1
2
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
S =1
2+
1
2S
S = 1
![Page 58: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/58.jpg)
The Sum
1
4+
1
8+
2
16+
3
32+
5
64+ . . . = 1
∞∑n=2
Fn−2
2n= 1
![Page 59: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/59.jpg)
The Sum
∞∑n=2
Fn−2
2n= 1
!!!
![Page 60: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/60.jpg)
The Sum
∞∑n=2
Fn−2
2n= 1!!!
![Page 61: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/61.jpg)
The Miracle
∑∞n=2
Fn−22n = 1!!!
THIS IS A MIRACLE!!!
![Page 62: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/62.jpg)
The Miracle
∑∞n=2
Fn−22n = 1!!!
THIS IS A MIRACLE!!!
![Page 63: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/63.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged. (And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 64: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/64.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged.
(And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 65: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/65.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged. (And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 66: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/66.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged. (And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 67: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/67.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged. (And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 68: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/68.jpg)
The Miracle
DefinitionA miracle is the simultaneous occurrence of two or morezero-probability events.
I The series converged. (And I could prove it!)
I We could find the value to which it converged.
I That value was rational.
I That value was the simplest possible rational.
![Page 69: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/69.jpg)
The Miracle
![Page 70: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/70.jpg)
The Very Good Reason
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
x g(x) =∞∑k=0
Fkxk+1 = x + x2 + 2x3 + 3x4 + . . .
x2g(x) =∞∑k=0
Fkxk+2 = x2 + x3 + 2x4 + . . .
So,g(x)− xg(x)− x2g(x) = 1
g(x) =1
1− x − x2
![Page 71: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/71.jpg)
The Very Good Reason
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
x g(x) =∞∑k=0
Fkxk+1 = x + x2 + 2x3 + 3x4 + . . .
x2g(x) =∞∑k=0
Fkxk+2 = x2 + x3 + 2x4 + . . .
So,g(x)− xg(x)− x2g(x) = 1
g(x) =1
1− x − x2
![Page 72: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/72.jpg)
The Very Good Reason
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
x g(x) =∞∑k=0
Fkxk+1 = x + x2 + 2x3 + 3x4 + . . .
x2g(x) =∞∑k=0
Fkxk+2 = x2 + x3 + 2x4 + . . .
So,g(x)− xg(x)− x2g(x) = 1
g(x) =1
1− x − x2
![Page 73: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/73.jpg)
The Very Good Reason
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
x g(x) =∞∑k=0
Fkxk+1 = x + x2 + 2x3 + 3x4 + . . .
x2g(x) =∞∑k=0
Fkxk+2 = x2 + x3 + 2x4 + . . .
So,g(x)− xg(x)− x2g(x) = 1
g(x) =1
1− x − x2
![Page 74: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/74.jpg)
The Very Good Reason
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
x g(x) =∞∑k=0
Fkxk+1 = x + x2 + 2x3 + 3x4 + . . .
x2g(x) =∞∑k=0
Fkxk+2 = x2 + x3 + 2x4 + . . .
So,g(x)− xg(x)− x2g(x) = 1
g(x) =1
1− x − x2
![Page 75: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/75.jpg)
The Very Good Reason
g(x) =1
1− x − x2
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
g
(1
2
)=∞∑k=0
Fk
(1
2
)k
= 1+1
2+2
(1
2
)2
+3
(1
2
)3
+5
(1
2
)4
+. . .
= 1 +1
2+
2
4+
3
8+
5
16+ . . .
= 4
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
![Page 76: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/76.jpg)
The Very Good Reason
g(x) =1
1− x − x2
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
g
(1
2
)=∞∑k=0
Fk
(1
2
)k
= 1+1
2+2
(1
2
)2
+3
(1
2
)3
+5
(1
2
)4
+. . .
= 1 +1
2+
2
4+
3
8+
5
16+ . . .
= 4
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
![Page 77: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/77.jpg)
The Very Good Reason
g(x) =1
1− x − x2
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
g
(1
2
)=∞∑k=0
Fk
(1
2
)k
= 1+1
2+2
(1
2
)2
+3
(1
2
)3
+5
(1
2
)4
+. . .
= 1 +1
2+
2
4+
3
8+
5
16+ . . .
= 4
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
![Page 78: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/78.jpg)
The Very Good Reason
g(x) =1
1− x − x2
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
g
(1
2
)=∞∑k=0
Fk
(1
2
)k
= 1+1
2+2
(1
2
)2
+3
(1
2
)3
+5
(1
2
)4
+. . .
= 1 +1
2+
2
4+
3
8+
5
16+ . . .
= 4
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
![Page 79: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/79.jpg)
The Very Good Reason
g(x) =1
1− x − x2
g(x) =∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
g
(1
2
)=∞∑k=0
Fk
(1
2
)k
= 1+1
2+2
(1
2
)2
+3
(1
2
)3
+5
(1
2
)4
+. . .
= 1 +1
2+
2
4+
3
8+
5
16+ . . .
= 4
(1
4+
1
8+
2
16+
3
32+
5
64+ . . .
![Page 80: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/80.jpg)
Technicalities
∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
This is the Taylor series about 0 for 11−x−x2 .
The interval of convergence is(−1
λ ,1λ
), where λ = 1+
√5
2 , i.e. thegolden mean.
(NB The radius of convergence, 1λ , is approximately .618, so 1
2 iscomfortably inside.)
![Page 81: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/81.jpg)
Technicalities
∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
This is the Taylor series about 0 for 11−x−x2 .
The interval of convergence is(−1
λ ,1λ
), where λ = 1+
√5
2 , i.e. thegolden mean.
(NB The radius of convergence, 1λ , is approximately .618, so 1
2 iscomfortably inside.)
![Page 82: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/82.jpg)
Technicalities
∞∑k=0
Fkxk = 1 + x + 2x2 + 3x3 + 5x4 + . . .
This is the Taylor series about 0 for 11−x−x2 .
The interval of convergence is(−1
λ ,1λ
), where λ = 1+
√5
2 , i.e. thegolden mean.
(NB The radius of convergence, 1λ , is approximately .618, so 1
2 iscomfortably inside.)
![Page 83: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/83.jpg)
Two Great Tastes That Taste Great Together
The function, g(x) = 11−x−x2 , is called the generating function for
the Fibonacci numbers.
![Page 84: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/84.jpg)
Turning the PageRewrite g(x) using partial fractions:
1
1− x − x2=
A
1− λx+
B
1 + (λ− 1)x
Solve for A and B:
A =λ√5
B =λ− 1√
5
By the known formula for geometric series:
A
1− λx=∞∑k=0
A (λx)k
Similarly,
B
1 + (λ− 1)x=∞∑k=0
B(−1)k ((λ− 1)x)k
![Page 85: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/85.jpg)
Turning the PageRewrite g(x) using partial fractions:
1
1− x − x2=
A
1− λx+
B
1 + (λ− 1)x
Solve for A and B:
A =λ√5
B =λ− 1√
5
By the known formula for geometric series:
A
1− λx=∞∑k=0
A (λx)k
Similarly,
B
1 + (λ− 1)x=∞∑k=0
B(−1)k ((λ− 1)x)k
![Page 86: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/86.jpg)
Turning the PageRewrite g(x) using partial fractions:
1
1− x − x2=
A
1− λx+
B
1 + (λ− 1)x
Solve for A and B:
A =λ√5
B =λ− 1√
5
By the known formula for geometric series:
A
1− λx=∞∑k=0
A (λx)k
Similarly,
B
1 + (λ− 1)x=∞∑k=0
B(−1)k ((λ− 1)x)k
![Page 87: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/87.jpg)
Turning the PageRewrite g(x) using partial fractions:
1
1− x − x2=
A
1− λx+
B
1 + (λ− 1)x
Solve for A and B:
A =λ√5
B =λ− 1√
5
By the known formula for geometric series:
A
1− λx=∞∑k=0
A (λx)k
Similarly,
B
1 + (λ− 1)x=∞∑k=0
B(−1)k ((λ− 1)x)k
![Page 88: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/88.jpg)
Never in a Million Years
1
1− x − x2=∞∑k=0
A (λx)k +∞∑k=0
B(−1)k ((λ− 1)x)k
Recalling the values of A and B,
∞∑k=0
Fkxk =∞∑k=0
[λk+1
√5
+(−1)k(λ− 1)k+1
√5
]xk
Power series expansions are unique! So,
Fk =λk+1
√5
+(−1)k(λ− 1)k+1
√5
![Page 89: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/89.jpg)
Never in a Million Years
1
1− x − x2=∞∑k=0
A (λx)k +∞∑k=0
B(−1)k ((λ− 1)x)k
Recalling the values of A and B,
∞∑k=0
Fkxk =∞∑k=0
[λk+1
√5
+(−1)k(λ− 1)k+1
√5
]xk
Power series expansions are unique! So,
Fk =λk+1
√5
+(−1)k(λ− 1)k+1
√5
![Page 90: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/90.jpg)
Never in a Million Years
1
1− x − x2=∞∑k=0
A (λx)k +∞∑k=0
B(−1)k ((λ− 1)x)k
Recalling the values of A and B,
∞∑k=0
Fkxk =∞∑k=0
[λk+1
√5
+(−1)k(λ− 1)k+1
√5
]xk
Power series expansions are unique! So,
Fk =λk+1
√5
+(−1)k(λ− 1)k+1
√5
![Page 91: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/91.jpg)
Be wise, generalize!
Heads Numerators Generatingand Recursion Function
One 1, 1, 1, 1, . . . 11−x
an = an−1Two 1, 1, 2, 3, . . . 1
1−x−x2an = an−1 + an−2
Three 1, 1, 2, 4, 7, . . . 11−x−x2−x3
an = an−1 + an−2 + an−3Four 1, 1, 2, 4, 8, 15, . . . 1
1−x−x2−x3−x4an = an−1 + an−2 + an−3 + an−4
And so on . . .
![Page 92: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/92.jpg)
“Suppose you had a three-sided coin?”
Heads Numerators Generatingand Recursion Function
One 1, 2, 4, 8, . . . 11−2x
an = 2an−1Two 1, 2, 6, 16, 48, . . . 1
1−2x−2x2an = 2(an−1 + an−2)
Three 1, 2, 6, 18, 52, 152, . . . 11−2x−2x2−2x3
an = 2(an−1 + an−2 + an−3)Four 1, 2, 6, 18, 54, 160, . . . 1
1−2x−2x2−2x3−2x4an = 2(an−1 + an−2 + an−3 + an−4)
Homework: Do the n consecutive heads from an m-sided coinproblem.
![Page 93: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/93.jpg)
Fibonacci Fractions
Do the long division problem:
10, 000
9, 899=
1.010203050813213455 . . .
What just happened?
g
(1
100
)= 1 +
1
100+
2
(100)2+
3
(100)3+ . . .
= 1 + .01 + .0002 + .000003 + . . .
g
(1
1000
)=
1, 000, 000
998, 999
![Page 94: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/94.jpg)
Fibonacci Fractions
Do the long division problem:
10, 000
9, 899= 1.010203050813213455 . . .
What just happened?
g
(1
100
)= 1 +
1
100+
2
(100)2+
3
(100)3+ . . .
= 1 + .01 + .0002 + .000003 + . . .
g
(1
1000
)=
1, 000, 000
998, 999
![Page 95: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/95.jpg)
Fibonacci Fractions
Do the long division problem:
10, 000
9, 899= 1.010203050813213455 . . .
What just happened?
g
(1
100
)= 1 +
1
100+
2
(100)2+
3
(100)3+ . . .
= 1 + .01 + .0002 + .000003 + . . .
g
(1
1000
)=
1, 000, 000
998, 999
![Page 96: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/96.jpg)
Fibonacci Fractions
Do the long division problem:
10, 000
9, 899= 1.010203050813213455 . . .
What just happened?
g
(1
100
)= 1 +
1
100+
2
(100)2+
3
(100)3+ . . .
= 1 + .01 + .0002 + .000003 + . . .
g
(1
1000
)=
1, 000, 000
998, 999
![Page 97: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/97.jpg)
Multiplying Weirdness
Recall
Fk =1√5
[λk+1 + (−1)k(λ− 1)k+1
]
Here’s a picture of those fractional parts:
Pictured is the fractional part of 1√5λk , k = 1, 2, . . . , 15.
![Page 98: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/98.jpg)
Multiplying Weirdness
Recall
Fk =1√5
[λk+1 + (−1)k(λ− 1)k+1
]Here’s a picture of those fractional parts:
Pictured is the fractional part of 1√5λk , k = 1, 2, . . . , 15.
![Page 99: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/99.jpg)
Your Real Homework
The picture for the fractional part of 1.5n for n = 10 . . . 80. (Thisis what we expect to see.)
![Page 100: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/100.jpg)
Your Real Homework
Conjecture
There are not very many real numbers, γ, that have the propertythat there exists a constant C so that the sequence consisting ofthe fractional parts of Cγn, n = 1, 2, . . . has only finitely manylimit points.
Not very many = measure zero= first category= nowhere dense
![Page 101: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/101.jpg)
Your Real Homework
Conjecture
There are not very many real numbers, γ, that have the propertythat there exists a constant C so that the sequence consisting ofthe fractional parts of Cγn, n = 1, 2, . . . has only finitely manylimit points.
Not very many = measure zero= first category= nowhere dense
![Page 102: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/102.jpg)
Your Real Homework: Hints
The number 1 +√
2 = p is even better than the golden mean.
p1 ≈ 2.41421
p2 ≈ 5.82842
p3 ≈ 14.07106
. . .
p8 ≈ 1153.99913
p9 ≈ 2786.00035
p10 ≈ 6725.99985
![Page 103: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/103.jpg)
Your Real Homework: Hints
p1 = 1 +√
2
p2 = 3 + 2√
2
p3 = 7 + 5√
2
. . .
p8 = 577 + 408√
2
p9 = 1393 + 985√
2
p10 = 3363 + 2378√
2
![Page 104: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/104.jpg)
A Final Word
Thanks to the local organizers: Adam Coffman, Rob Merkovskyand Marc Lipman.
Thanks to IUPU–Fort Waye.
Thank you for your kind attention.
![Page 105: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/105.jpg)
A Final Word
Thanks to the local organizers: Adam Coffman, Rob Merkovskyand Marc Lipman.
Thanks to IUPU–Fort Waye.
Thank you for your kind attention.
![Page 106: Two Heads Are Better Than None Or, Me and the Fibonaccisskennedy/KSTwoHeads.pdf · The Problem The Game: Flip a fair coin repeatedly until two consecutive heads appear, stop. The](https://reader034.vdocument.in/reader034/viewer/2022042320/5f0a19727e708231d42a032c/html5/thumbnails/106.jpg)
A Final Word
Thanks to the local organizers: Adam Coffman, Rob Merkovskyand Marc Lipman.
Thanks to IUPU–Fort Waye.
Thank you for your kind attention.