Conditional Expectation given n?

Question

At a 24-hour movie theater, customers arrive at a rate of 10 customers per hour. Any given customer will independently buy a ticket for one of the following movie genres with corresponding probabilities:

-Action, 0.4 -Romance, 0.3 -Horror, 0.2 -Comedy, 0.1

Given there were 300 customers who bought tickets today, find the expected value and distribution (i.e., X~Binom(x, y)) of the number of romance tickets bought.

I can't figure out whether this is a conditional expectation problem or simply n*p = 300*0.3, however, this also arises an issue of how I would give the distribution.

Any help would be appreciated!

Answer 1

$X \sim \mathsf{Binom}(n = 300, p = .03).$ and, as you say $E(X) = np = 300(.3) = 90.$

Plot using R:

x = 50:150;  pdf = dbinom(x, 300, .3)
plot(x, pdf, type="h", lwd=2, main="BINOM(300,0.3)")
 abline(h=0, col="green2")
 abline(v=90, col="red", lty="dotted",lwd=2)