[pudley4]

9/47 = the probability of hitting on the turn.

9/46 = the probability of hitting on the river if you miss on the turn. You won't always miss on the turn (you'll only miss 38/47 times), so you can't just add 9/46, you have to add (9/46)*(38/47) - this is the probability of missing the turn then hitting the river.

There are 4 possibilities:

A - You miss on the turn and the river (38/47 * 37/46)
B - You miss on the turn and hit on the river (38/47 * 9/46)
C - You hit on the turn and miss on the river (9/47 * 38/46)
D - You hit on the turn and hit on the river (9/47 * 8/46)

Notice that B, C, and D show all the possible ways you can win the hand. You can calculate each of these 3 separately, then add them together. A shorter way is to notice that C and D added together equal the probability you hit on the turn. So you can calculate B, then add 9/47. This is the same as the beginning of my post.

Or, you can notice the easiest way - since A shows all the ways you can miss, you can calculate A, then subtract the result from 1 (since all the probabilities added together must equal 1)