Question

A high school baseball player has a 0.159 batting average. In one game, he gets 9 at bats. What is the probability he will get at least 7 hits in the game?

Answer 1

0.0068% probability he will get at least 7 hits in the game

Explanation:

For each at bat, there are only two possible outcomes. Either he gets a hit, or he does not. The probability of getting a hit in an at-bat is independent of other at-bats. So we use the binomial probability distribution 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.

A high school baseball player has a 0.159 batting average.

This means that in each at-bat, the probability of getting a hit is 0.159.

9 at bats.

This means that
`n = 9`

What is the probability he will get at least 7 hits in the game?

`P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9)`

In which

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

`P(X = 7) = C_(9,7).(0.159)^(7).(0.841)^(2) = 0.000065`

`P(X = 8) = C_(9,8).(0.159)^(8).(0.841)^(1) = 0.000003`

`P(X = 9) = C_(9,9).(0.159)^(9).(0.841)^(0) \cong 0`

`P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) = 0.000065 + 0.000003 = 0.000068`

0.0068% probability he will get at least 7 hits in the game