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
Geoff K.
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)#
#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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