I have a few basic probability questions, concerning the following question:
An internet survey estimates that, when given the choice between David letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Three late night TV watchers are randomly selected and asked which of the two talk show hosts they prefer. Find the probability distribution of Y, the number of viewers in the sample who prefer Leno.
The answer is:
P(Y=0) = (0.48)^3
P(Y=1) = 3 * (0.48)^2 (0.52)
P(Y=2) = 3 * (0.48) (0.52)^2
P(Y=3) = (0.52)^3
I understand the solution mostly, except for where the 3 comes from in P(Y=1) and P(Y=2). Is it because there are 3 ways of getting 1 and 2 viewers who prefer Leno in the sample space? If so, is there a mathematical way of calculating this or is it simply writing out the entire sample size/combinations, and then counting?
Thanks so much.
The probability that $k=0,\,1,\,2,\,3$ viewers prefer Leno is a binomial distribution, and is given by
$$P(Y=k)={3\choose k}p^k(1-p)^{3-k}$$
where $p=0.52$, and
$${3\choose k}=\frac{3!}{k!(3-k)!}$$.