Unizor - Probability - Advanced Problems 3

Problem A
M passengers enter a bus at a bus stop in the beginning of its route. There are S stops on the bus route (not counting the beginning stop) where these passengers can exit.
Let's assume that the probability of any passenger to exit on any stop is the same, that is, they exit completely randomly.
Assuming that the number of passengers M is less than the number of bus stops S and no new passengers are entering a bus, what is the probability of
(a) all of them exit at different stops?
(b) all of them exit at the same stop?
(c) one passenger exits at each sequential stop until all of them are exited, so that Mth passenger exits at the Mth stop?

Answer:
(a) C(S,M)/S^M;
(b) 1/S^(M−1);
(c) M!/S^M.

Problem B
Six cards are randomly pulled from a standard deck of 52 cards.
What is the probability of having all four suits represented among these six cards?

Answer:
P = {C(4,1)·C(13,3)·[C(13,1)]^3 + C(4,2)·[C(13,2)]^2·[C(13,1)]^2} / C(52,6)