FUNDAMENTALS
Card probability without replacement
Calculate outs exactly using complementary events, conditional probabilities and small sample spaces.
Start with the sample space
On the turn, a player knows two hole cards and four community cards. There are 46 unseen cards. If exactly nine cards are clean winning outs, the probability of hitting one on the river is 9 / 46, about 19.565%.
An out is only clean if it really gives the desired result. Flush-completion probability and showdown-winning probability are not interchangeable.
For multiple draws, count misses
On the flop, 47 cards are unseen. With nine outs, 38 miss. If the turn misses, 46 cards remain and 37 miss.
The probability of at least one out is one minus the probability of missing both. This is approximately 34.968%, assuming the set of outs remains valid.
P(at least one hit) = 1 − (38 / 47)(37 / 46)
Check an answer with a simpler case
For a pocket pair, condition on the first card. Three of the remaining 51 cards have its rank, so the probability is 3 / 51, about 5.882%.
The ability to choose a convenient conditioning event is useful in probability interviews. It often replaces a long combinatorial calculation with one short argument.