Among 25 Presidential candidates 11 think global warming is

Among 25 Presidential candidates, 11 think global warming is a myth, 8 believe that global warming is real, and the rest have no opinion (\"I\'m not a scientist\"). A newspaper interviews a random sample of 5 of the candidates. What is the probability that

(a) all 5 think global warming is a myth;

(b) all 5 share the same position (i.e., all think it is a myth, all believe it is real, or all have no opinion);

(c) 3 share a common position and 2 share a common position which is different from the other 3 (for example, 3 believe it is a myth and 2 have no opinion, or 3 have no opinion and 2 believe it is real). [Hint: First figure out how many different choices of the two different positions there are, then the number of possible choices for each of these.]

Solution

a)
Picking a 5 People who think it is a myth = 11 C 5 / 25 C 5 = 0.0086
b)
P(all think it is a myth, all believe it is real, or all have no opinion) = 11 C 5 / 25 C 5 + 8 C 4 / 25 C 5 + 6 C 5 / 25 C 5 = 0.0086 + 0.00105 + 0.000112 = 0.009762
c)
different choices of the two different positions = P(3 Myth) * P(2 No Opinion) + P(3 Myth) * P(2 Real) + P(3 Real) * P(2 No Opinion) + P(3 Real) * P(2 Myth) + P(3 No Opinion) * P(2 Real) + P(3 No Opinion) * P(2 Myth) = [ 11 C 3 * 6 C 2 + 11 C 3 * 8 C 2 + 8 C 3 * 6 C 2 + 8 C 3 * 11 C 2 + 6 C 3 * 11 C 2 + 6 C 3 * 8 C 2 ] / 25 C 5 = 0.238565