a woman standing on a bridge overlooking a major insterstate wants to know how likely it is that the next 20 out of 50 cars that pass under the bridge are red cars, if she has observed in the past that the probability than any given car on that interstate highway is red is approximately 0.28.

Question

a woman standing on a bridge overlooking a major insterstate wants to know how likely it is that the next 20 out of 50 cars that pass under the bridge are red cars, if she has observed in the past that the probability than any given car on that interstate highway is red is approximately 0.28.

What is the probability?

Answer 1

Your question is worded in a confusing way. It sounds like the next 20 cars IN A ROW would have to be red. That would be a very low probability, (0.28)^20 = 8.8*10^-12, and the 30 cars seen after that would not matter.

Don't you mean that you want the probability that exactly 20 of the next 50 will be red?
That probability is
P(20) = (0.28)^20*(0.72)^30^50!/[30!*20!]
= 0.0217

The most likely number of red cars out of 50 would be 14. That probability is
P(14) = (0.28)^14*(0.72)^36* 50!/[14!*36!] = 0.1248
=