Of the 15 members on a Senate committee 10 plan to vote yes

Of the 15 members on a Senate committee, 10 plan to vote “yes” and 5 plan to vote “no” on an
important issue. A reporter attempts to predict that outcome of the vote by questioning six of the senators. Find the probability that this sample is precisely representative of the final vote. That is, find the probability that four of the six senators questioned plan to vote “yes”.

Solution

n=15, p=10/15=.6667, q=5/15=.3333. It follows binomial distrubution P(X=x)=(15cx)(.6667)^x(.3333)^(n-x), then P(X=6)=(15c6)(.6667)^6(.3333)^9=.0223, P(X=5)=(15c5)(.6667)^5(.3333)^10=.0067. P(X<=4)=1-[P(X=6)+P(X=5)]=1-[.0223+.0067]=1-.029=.971.