Three students take a test with 50% of passing. Suppose that the test is opened every month. If a student fails, he will take the next test until he passes. Calculate the expected number of months that all of three students pass the test.
At the first sight, I thought this question is simple that I call X is the number of months that all of three students pass the test. Then E(X) = $\Sigma_{x=1}^\infty x*f(x)$.However, it is very exhausted to calculate $f(x)$.
I also try to calculate separate probability on each student and use joint probability to calculate E(X) but I got stucked too.
The probability, $Q(j)$, that exactly $j$ months are needed for the students to all pass is the probability that every student takes at most $j$ months minus the probability that every student takes at most $j-1$ months.
Since the students’ results are independent we can write this as:
$Q(j) = P(j)^3 - P(j-1)^3$
where $P(j)$ is the probability that a student passes on attempt $j$ or before.
$P(j) = 1 - 1/2^j$
Substituting and simplifying, we get
$Q(j) = 3/2^j - 9/4^j + 7/8^j$
As a check, we can sum $Q(j)$ to infinity. Summing the three geometric series separately, we get the value $3-3+1 = 1$, which is correct.
But we want the expectation, i.e. the sum of $j*Q(j)$ to infinity.
This turns out to be $6-4+8/7 = 22/7$ which is a plausible value and I guess is unconnected with $\pi$ here!