chapter 15 probability models
DESCRIPTION
Chapter 15 Probability Models. The Equally Likely Approach (also called the Classical Approach). Assigning Probabilities. If an experiment has N outcomes, then each outcome has probability 1/N of occurring If an event A 1 has n 1 outcomes, then P(A 1 ) = n 1 /N. Dice - PowerPoint PPT PresentationTRANSCRIPT
Assigning ProbabilitiesIf an experiment has N outcomes,
then each outcome has probability 1/N of occurring
If an event A1 has n1 outcomes, thenP(A1) = n1/N
DiceYou toss two dice. What is the probability of the outcomes summing to 5?
There are 36 possible outcomes in S, all equally likely (given fair dice).
Thus, the probability of any one of them is 1/36.
P(the roll of two dice sums to 5) =
P(1,4) + P(2,3) + P(3,2) + P(4,1) = 4 / 36 = 0.111
This is S:
{(1,1), (1,2), (1,3), ……etc.}
Counting in “Either-Or” Situations• NCAA Basketball Tournament, 68
teams: how many ways can the “bracket” be filled out?
1. How many games?2. 2 choices for each game3. Number of ways to fill out the bracket:
267 = 1.5 × 1020
• Earth pop. about 6 billion; everyone fills out 100 million different brackets
• Chances of getting all games correct is about 1 in 1,000
Counting ExamplePollsters minimize lead-in effect by
rearranging the order of the questions on a survey
If Gallup has a 5-question survey, how many different versions of the survey are required if all possible arrangements of the questions are included?
SolutionThere are 5 possible choices for the
first question, 4 remaining questions for the second question, 3 choices for the third question, 2 choices for the fourth question, and 1 choice for the fifth question.
The number of possible arrangements is therefore
5 4 3 2 1 = 120
Efficient Methods for Counting Outcomes
Factorial Notation:n!=12 … n
Examples1!=1; 2!=12=2; 3!= 123=6; 4!
=24;5!=120;Special definition: 0!=1
Factorials with calculators and Excel
Calculator: non-graphing: x ! (second function)graphing: bottom p. 9 T I Calculator Commands(math button)
Excel:Insert function: Math and Trig category, FACT function
Factorial Examples20! = 2.43 x 1018
1,000,000 seconds?About 11.5 days1,000,000,000 seconds?About 31 years31 years = 109 seconds1018 = 109 x 109
20! is roughly the age (according to some) of the universe in seconds
Permutations
A B C D EHow many ways can we choose 2
letters from the above 5, without replacement, when the order in which we choose the letters is important?
5 4 = 20
Permutations with calculator and Excel
Calculatornon-graphing: nPr
Graphingp. 9 of T I Calculator Commands(math button)
ExcelInsert function: Statistical, Permut
Combinations
A B C D EHow many ways can we choose 2
letters from the above 5, without replacement, when the order in which we choose the letters is not important?
5 4 = 20 when order importantDivide by 2: (5 4)/2 = 10 ways
ST 305 Powerball Lottery
From the numbers 1 through 20,choose 6 different numbers.
Write them on a piece of paper.
Chances of Winning?
760,38!6)!620(
!20
ies?possibilit ofNumber important.not order t,replacemen
without 20, from numbers 6 Choose
620206
C
Example: Illinois State Lottery
balls) pong pingmillion 16.5 house, ft (1200
months) 10in second 1about (
165,827,25!6!48
!54importantnot order t;replacemen
withoutnumbers 54 from numbers 6 Choose
2
654 C
North Carolina Powerball Lottery
Prior to Jan. 1, 2009 After Jan. 1, 2009:
55! 3, 478,7615!50!
:42! 42
1!41!3,478,761*42146,107,962
5 from 1- 55
1 from 1- 42 (p'ball #)
:59! 5,006,386
5!54!:
39! 391!38!5,006,386*39195,249,054
5 from 1- 59
1 from 1- 39 (p'ball #)
Most recent change: powerball number is from 1 to 35http://www.nc-educationlottery.org/faq_powerball.aspx#43
The Forrest Gump Visualization of Your Lottery Chances
How large is 195,249,054?$1 bill and $100 bill both 6” in length
10,560 bills = 1 mileLet’s start with 195,249,053 $1 bills
and one $100 bill …… and take a long walk, putting down
bills end-to-end as we go
Chances of Winning NC Powerball Lottery?
Remember: one of the bills you put down is a $100 bill; all others are $1 bills.
Put on a blindfold and begin walking along the trail of bills.
Your chance of winning the lottery is the same as your chance of selecting the single $100 bill if you stop at a random location along the trail and pick up a bill .
More ChangesAfter Jan. 1, 2009 After Jan. 1, 2012
http://www.nc-educationlottery.org/powerball_how-to-play.aspx
:59! 5,006,386
5!54!:
39! 391!38!5,006,386*39195,249,054
5 from 1- 59
1 from 1- 39 (p'ball #)