applied math 40s march 3, 2008
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Permutations of Non-Distinguishable Objects ... like twins. ;-)
My Lovely Twins
but for real this time ...
(a) In how many ways can 4 English books and 3 French books be arranged in a row on a shelf?
(b) In how many of these ways will the French books be together?
HOMEWORK
If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?
Permutations of non-distinguishable objects ...
The number of ways to arrange n objects that contain k, k , k , ... sets of non-distinguishable objects is given by:
Find the number of different "words" that can be made by rearranging the letters in the word:
Examples:
(a) BOOK (b) MISSISSIPPI
32
B K
B K
B K
K B
B K
K B
KB
K B
BK
K B
K B
B K
KB
KB
KBKB
K B
K B
KB
K B
KB
KB
K B
K B
If a fair coin is tossed 4 times, what is the probability of obtaining exactly 2 heads?
All the letters of the word MANITOBA are arranged at random in a row. How many ways can this be done?
What is the probability that this random arrangement will have the two A’s next to each other?
How many arrangements will have the two A’s next to each other?
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(a) How can this experiment be simulated on your calculator?What would your type into the calculator?
(b) Carry out the experiment, on your calculator a large number of times. Indicate the number of times your simulation ran the experiment and record the result (the experimental probability).
(c) What is the solution to this problem using theoretical probability? A tree diagram will help. Hint: It will have six steps.
(d) Are your answers to (b) and (c) the same? Did you expect them to be? Explain.
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(a) How can this experiment be simulated on your calculator?What would your type into the calculator?
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(b) Carry out the experiment, on your calculator a large number of times. Indicate the number of times your simulation ran the experiment and record the result (the experimental probability).
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(b) Carry out the experiment, on your calculator a large number of times. Indicate the number of times your simulation ran the experiment and record the result (the experimental probability).
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(c) What is the solution to this problem using theoretical probability? A tree diagram will help. Hint: It will have six steps.
Give a Worm a Chance!
A food inspector at a grocery store discovers that 25% (that's 1 in 4) of all the apples in the store have worms. Lucky Luke, a shopper, buys six apples. He randomly selects the apples from the bin without checking them. What is the probability that none of the six apples he buys has worms?
(d) Are your answers to (b) and (c) the same? Did you expect them to be? Explain.
NO (at least I hope you didn't)
Repeating the experiment many many many times would result in the experimental probability getting closer and closer to the theoretical probability.
NO
If 8 books are arranged on a shelf, what is the probability that 3 particular books are together?
HOMEWORK
(a) In how many ways can the letters of the word GEOMETRY be arranged so that vowels and consonants alternate?
(b) In how many of these ways is Y the last letter?
(c) If one of these "words" is randomly selected, what is the probability that Y is the last letter?
HOMEWORK
Suppose that, when you go home from school, you like to take as great a variety of routes as possible, and that you are equally likely to take any possible route. You will walk only east or south.
How many ways can you go from the school to home?
What is the probability that you will walk past the post office on your way home?
HOMEWORK
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