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

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

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Answer 1 · 2017-01-28T01:51:33

The probability is exactly $23.04 %$ $23.04%$ .

Explanation:

This question calls for our old friend the Binomial distribution.

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

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

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

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

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

1 Answer

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