Odds Question?

[Kenser]

I know this has been asked before but here goes again. If you have 2-suited and 1 of the same suite appears on the flop, what are the odds of the same suite appearing on the turn and river? Is it about 23 to 1? I see people drawing to this all the time with no other outs!

[Mike Haven]

here's one way of working it out from first principles:

you have seen 5 cards

3 are your suit

there are 47 unseen cards, including 10 of your suit

so the turn will be your suit 10 times in every 47 turn deals

for ease of the next stage of calculations, multiply these figures by 46

so the turn will be your suit 460 times in every 2162 turn deals

and of these 460 successful turn deals the river will also be your suit 9 times in every 46 river deals, for similar reasons as above

or 90 times in the 460

so that means that both will be your suit 90 times in 2162 turn plus river deals

or once in every 24

so the odds against receiving a runner runner flush are 23 to 1

[Lin Sherman]

Remember the number 1081. This is the total possible combinations of the last two cards when you've already seen 5 of them (i.e., your hand and the flop). The question is how many of these 2-card combinations contain two of the suit in question.

If you have two of the flush cards and the board has one that leaves 10 in the deck. There are 10 x 9 / 2 = 45 ways to combine these 10 cards into 2-card combinations. So the probability is 45/1081 = ~1/24 = 23-to-1 odds.

Lin