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The probability that Barry Bonds hits a home run on any given at-bat is 0.16, and each at-bat is independent.

Part A: What is the probability that the next home run will be on his fifth at-bat? (5 points)

Part B: What is the expected number of at-bats until the next home run? (5 points)

1 Answer

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Answer:

a) 0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

b) The expected number of at-bats until the next home run is 6.25.

Explanation:

For each at bat, there are two possible outcomes. Either it is a home run, or it is not. The probability of an at bat resulting in a home run is independent of any other at-bat, which means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.


P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)

In which
C_(n,x) is the number of different combinations of x objects from a set of n elements, given by the following formula.


C_(n,x) = (n!)/(x!(n-x)!)

And p is the probability of X happening.

The probability that Barry Bonds hits a home run on any given at-bat is 0.16

This means that
p = 0.16

Part A: What is the probability that the next home run will be on his fifth at-bat?

0 on his next 4(P(X = 0) when n = 4)

Home run on his 5th at-bat, with 0.16 probability. So


P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)


P(X = 0) = C_(4,0).(0.16)^(0).(0.84)^(4) = 0.49787136

0.49787136 *0.16 = 0.0797.

0.0797 = 7.97% probability that the next home run will be on his fifth at-bat.

Part B: What is the expected number of at-bats until the next home run?

The expected number of trials for n successes is given by:


E = (n)/(p)

In this question,
n = 1, p = 0.16. So


E = (1)/(0.16) = 6.25

The expected number of at-bats until the next home run is 6.25.

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