Download 6.1 - 6.2 Applications

AP Statistics – Probability Simulations and Probability Models Applications
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Name:
Pau Kerr is a card shark who likes to play the following game. Draw 2 cards from a standard deck of 52 cards. If at least
one of the cards is a heart, you win $1. If neither card is a heart, you lose $1.
a. Describe a simulation that would show a correspondence between random numbers and possible outcomes in this
game.
b. Simulate playing this game for 25 rounds using random numbers (calculator or table). How many of the 25 games
did you win?
c. Combine your data with 3 other students so that you have a total of 100 simulations. How many times out of 100
did you beat Pau Kerr? Is this a fair game?
Suppose a major league baseball player has a current batting average of .320 [AVG = number of hits/number of at bats].
a. Describe an assignment of random numbers to possible results in order to simulate the player’s next 20 at bats.
b. Carry out the simulation 20 times, and report your results. What is the relative frequency of at bats in which the
player gets a hit?
c. Compare your simulated experimental results for the next 20 at bats with the player’s actual batting average.
d. Combine your data with 3 other students. What has changed?
Brian and Corey play a game of “heads” (H) or “tails” (T). They toss a coin 4 times. Brian wins a dollar from Corey for each
H and pays Corey a dollar for each T [Brian wins or loses the difference between the number of H and the number of T – for
example, if a game results in one H and thee T, Brian loses $2]. Brian’s possible outcomes are {-4, -2, 0, 2, 4}. Assign
probabilities to these outcomes by playing the game 20 times [or simulating 20 times] and using the proportions of the
outcomes to estimate the probabilities. Combine your trials with those of other students to obtain long-run proportions
that are closer to the probabilities.
Probability is a measure of how likely an event is to occur. Match each statement about an event with one of the
probabilities that follow: 0
0.01
0.3
0.6
0.99
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a. This event is impossible. It can never occur.
b. This event is certain. It will occur on every trial of the random phenomenon.
c. This event is very unlikely, but it will occur once in a while in a long sequence of trials.
d. This event will occur more often than not.
A study of the home field advantage in baseball found that over the period from 1969 to 1989 the league champions won
63% of their home games. The two league champions meet in the World Series. Would you use the study results to assign
probability 0.63 to the event that the home team wins in a World Series game? Explain.
The probability of being dealt two pairs in a poker game is approximately 1/21. Explain in plain language what P(two-pair) =
1/21 means. Also, explain why it does NOT mean that in 21 deals you will get exactly one two-pair hand.
In each of the following situations, describe a sample space S for the random phenomenon.
a. A seed is planted in the ground. It either germinates or fails to grow.
b. A student enrolls in a statistics course and at the end of the semester receives a letter grade [solid grades – no +/-].
c. A basketball player shoots four free throws. You record the sequence of hits and misses.
d. A basketball player shoots four free throws. You record the number of baskets made.
For each of the following, use a tree diagram or the multiplication principle to determine the number of outcomes in the
sample space.
a. Suppose a county license tag has a four-digit number for identification. If any digit can occupy any of the four
positions, how many county license tags can you have?
b. If the county license tags in (a) do not allow duplicate digits, how many tags can you have?
c. Suppose the county license tags can have up to 4 digits. How many county license tags will this scheme allow?
Suppose you pick a card from a standard deck of 52 playing cards. In how many ways can the selected card be:
a. A red card?
b. A heart?
c. A queen and a heart?
d. A queen or a heart?
e. A queen that is not a heart?
10. Select a 1st year college student at random and ask what his or her academic rank was in high school. Here are the
probabilities, based on proportions from a large sample survey of 1 st year students:
Rank
Top 20%
Second 20%
Third 20%
Fourth 20%
Lowest 20%
Probability
.41
.23
.29
.06
.01
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What is the sum of these probabilities? Why should the sum have this value?
What is the probability that a randomly chosen 1st year college student was not in the top 20% of his or her high
school?
What is the probability that a 1st year student was in the top 40% in high school?
11. The first digits of numbers in legitimate financial records often follow a distribution known as Benford’s Law. Here is the
distribution:
First Digit
1
2
3
4
5
6
7
8
9
Probability .301
.176
.125
.097
.079
.067
.058
.051
.046
P(A) = P(first digit is 1) = .301
P(B) = P(first digit is 6or greater) = .222
P(C) = P(first digit is odd) = .609
D = first digits less than 4
Find the following:
a. P(D)
b. P(B U D)
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d.
e.
P(C ∩ D)
P(B ∩ C)
12. A general can plan a campaign to fight one major battle of three small battles. He believes that he has the probability of 0.6
of winning the large battle and a probability of 0.8 of winning each of the small battles. Victories or defeats in the small
battles are independent. The general must win either the large battle or all three small battles to win the campaign. Which
strategy should he choose?
13. An automobile manufacturer buys computer chips from the supplier. The supplier sends a shipment containing 5%
defective chips. Each chip chosen from this shipment has a probability of 0.05 of being defective, and each automobile uses
12 chips selected independently. What is the probability that all 12 chips in a car will work properly?
14. A string of holiday lights contains 20 lights. The lights are wired in series, so that if any light fails the whole string fails. Each
light has a probability of 0.02 of failing during a 3 year period. The lights fail independently of each other. What is the
probability that the string of lights will remain bright for 3 years?
15. An athlete suspected of having used steroids is given two tests that operate independently of each other. Test A has a
probability of 0.9 of being positive if steroids have been used. Test B has a probability of 0.8 of being positive if steroids
have been used. What is the probability that neither test is positive is steroids have been used?