A test consists of fifteen true/false questions. A student who forgot to study guesses randomly on every question. What is the probability that the student answers exactly five questions correctly?
+500 First, the number of ways to choose 5 questions out of 15 is equivalent to
\(15 \choose 5 \)= \(15!/(10!*5!) = 15*14*13*12*11/(5*4*3*2*1)\)
Canceling out top and bottom, we get:
\(3*7*13*11 = 3003\)
Next, we need the probability that he gets exactly five questions, which is:
\((1/2)^5\)
However, we haven't accounted for the next 10 questions which he needs to get wrong, which is a probability of
\((1/2)^{10}\)
Our answer is then:
\(3003*{1/2}^5*{1/2}^{10} = 3003/(2)^{15}\approx 0.0916442871\)
