Answer
(a) $0.0171$
(b) $≈ 0.902$
(c) $50$
Work Step by Step
(a) We can model this situation using a geometric distribution, where X is the number of calls it takes to get the first successful (i.e. connecting) call. Since the probability of success is $p = 0.02,$ we have:
$P(X = k) = (1-p)^{k-1} * p$
where $k$ is the number of calls it takes to get the first successful call. So, the probability that the first call that connects is your 10th call is:
$P(X = 10) = (1-0.02)^{10-1} * 0.02 = 0.0171$
Therefore, the probability that your first call that connects is your 10th call is approximately $0.0171.$
(b) We want to find $P(X > 5)$, which is the probability that it takes more than five calls to get a successful call. We can use the complement rule and calculate P(X ≤ 5) instead, and then subtract it from 1. So:
$P(X ≤ 5) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)$
$ = (1-0.02)^{1-1} * 0.02 + (1-0.02)^{2-1} * 0.02 + (1-0.02)^{3-1} * 0.02 + (1-0.02)^{4-1} * 0.02 + (1-0.02)^{5-1} * 0.02
≈ 0.098$
Therefore, the probability that it requires more than five calls for you to connect is:
$P(X > 5) = 1 - P(X ≤ 5) ≈ 0.902$
(c) The mean of a geometric distribution with parameter p is:
$E(X) = 1/p$
So, the mean number of calls needed to connect is:
$E(X) = 1/0.02 = 50$
Therefore, on average, you'll need to make 50 calls to connect. But don't worry, you can always sing along to the hold music while you wait!
