I have the following question:
If I hold a Jack and a nine, what is the chance of making a straight (five cards in a row) when the next three cards have been dealt?
My attempt at an answer:
There are three possible 5 card hands that make a straight:
7 8 9 T J
8 9 T J Q
9 T J Q K
There are 50 cards unseen, and there are 4 each of the 7's, 8's and 10's, so the possibility of making the first straight is the joint probability:
12/50 x 8/49 x 4/48 = 0.00327
What I can't work out is, are the probabilities for each of those 3 straights independent or not?
I was also trying to work this out using combinations, by the following method:
(4C1 x 4C1 x 4C1) / 50C3 = 4/1225 = 0.00327
and doing it that way, it's easier to see that there 64 possible combinations for the first straight and 64 combinations each for the other 2. So I'm pretty sure that this means that make one type of straight or another are independent events, but I'd be grateful if someone could maybe shed a bit more light on exactly why.