Each year, a campus society of 15 students is formed, composed of three groups: 4 Architecture students,
Question:
Each year, a campus society of 15 students is formed, composed of three groups: 4 Architecture students, 5 Botany students, and 6 Composition students. Each year, independently, a committee of 3 students is selected at random from among the 15 students. In any particular year, we say that Architecture wins if all 3 members of the committee are from Architecture, Botany wins if the three are from Botany, and Composition wins if the three are from Composition.
(a) Compute the probability that Botany wins next year.
(b) Compute the probability that one of the three groups wins next year.
(c) Let N be the number of years, starting from next year, up to (and including) the first year in which one of the three groups wins for the first time. Identify the distribution of N and compute EN.
(d) Compute the probability that the first group to ever win is Botany. Give the result as a reduced fraction.
(e) Let N be as in (c) and X the indicator of the event from (d) that the first group to ever win is Botany. Are N and X independent?