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

Jan 28, 2017

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)
$\textcolor{w h i t e}{P \left(H = 3\right)} = 10 \times {\left(0.4\right)}^{3} {\left(0.6\right)}^{2}$
$\textcolor{w h i t e}{P \left(H = 3\right)} = 10 \times 0.064 \times 0.36$
$\textcolor{w h i t e}{P \left(H = 3\right)} = 0.2304$

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

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