Question

A player's batting average in softball is .400. What is the probability he gets 3 hits in his next 5 at-bats?

Answer 1

The probability is exactly `23.04%`.

Explanation:

This question calls for our old friend the Binomial distribution.

If we let `H` be the number of hits in the next 5 at-bats, and we assume each at-bat is independent, then `H` has a Binomial distribution with `n=5` and `p=0.4`.

Why? Because the probability of a hit on a single at-bat is 0.4, and if the player gets `h` hits out of 5 tries, the number of sequences that produce those `h` hits is `""_5C_h`.

So we have `H ~ "Bin"(n=5, p=0.4)`.

From there, we just plug the given values for `n`, `p`, and `h` into the binomial formula to get

`P(H=h)=""_5C_h (0.4)^h(1-0.4)^(5-h)`
`P(H=3)=""_5C_3 (0.4)^3(0.6)^(5-3)`
`color(white)(P(H=3))=10 xx (0.4)^3(0.6)^2`
`color(white)(P(H=3))=10 xx 0.064 xx 0.36`
`color(white)(P(H=3))=0.2304`

So the probability of getting exactly 3 hits out of the next 5 at-bats is

`P(H=3)=0.2304" "=" "23.04%.`