Probability problem. Possibly involves combinatorics.
I'm a mature student undertaking a self-study A-Level course in order to go to university. I have been unable to figure this out by myself and have even had a private tutor stumped by it, so any help would be greatly appreciated.
It's from "Understanding Statistics" by Upton & Cook. Question 5g) 1) d). The correct answer is 613/648 according to the textbook.
In Ruritania all the cars are made by a single firm and vary only in their colouring. Six different colours are available. The same numbers of cars are painted in each of the six colours. Assuming that, when travelling on the roads, the colours of the cars occur in random order, determine the probability that:
"at least 8 cars pass before all 6 colours are encountered."
Re: Probability problem. Possibly involves combinatorics.
The first thing you have to do is come up with a distribution that corresponds to the actual problem/process.
In your problem this is a multivariable hyper-geometric distribution since you are sampling without replacement. However in the case that you have a large sample size, you can approximate this well with a multinomial distribution.
In your probability, you need to find out P(C1 > 0, C2 > 0, C3 > 0, ..., C6 > 0, N > 7).
Hint: With a multinomial, try finding the probability that less than 8 cars pass with all colors appearing once and find 1 - that probability.
Multinomial distribution - Wikipedia, the free encyclopedia
Remember that we are using a approximation if the number of cars is extremely large.