How to Solve Flop to River with Differing Outs?

[Lost Wages]

What is the general proceedure for using the combinations method to determine the probability of hitting either one or two of your outs when the number of outs is different on the flop and turn? For example, a set, which has 7 outs on the flop and 10 outs on the turn.

In "Hold'em's Odd(s) Book", Petriv solves this problem by adding the probability of making quads or a full house with the 7 flop outs to the probability of making a running pair.

Is there a more general way to determine the flop to river probabilities (hitting 1 out, hitting 2 outs, hitting 1 or 2 outs) if the flop outs are X and the turn outs are Y ?

Please show your work /images/graemlins/smile.gif,
Lost Wages

[Lost Wages]

OK, I've figured out how to do it using probability (I think).

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If you have X outs on the flop and Y outs on the turn then:

P(hitting 2 outs) = P(out on turn) * P(out on river)
= (X/47)*((Y-1)/46)

P(hitting exactly 1 out) = P(out on turn * blank on river) + P(blank on turn * out on river)
= ((X/47)*((46-(Y-1))/46)) + (((47-X)/47)*(Y/46))

P(hitting either 1 or 2 outs) = P(hitting 2 outs) + P(hitting exactly 1 out)

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Solving for the case of a set where X=7 and Y=10 yields:

P(hitting 2 outs) = 2.91%
P(hitting exactly 1 out) = 30.48%
P(hitting either 1 or 2 outs) = 33.39%

Lost Wages

[Bozeman]

For the set to fullhouse or quads problem, y=10 only if you miss on the turn, but you essentially never need to hit twice, however.

Craig

[Lost Wages]

y=10 only if you miss on the turn

I thought I had taken care of that in the equations by using (Y-1) where appropriate. Did I screw it up? Also, I was concerned about the case where you make quads on the turn. Then you have zero outs so the "hitting 2 outs" make no sense.

Lost Wages

[Bozeman]

My point was that if you hit on the turn (say qq w qt77), you don't have the new card on board that can pair. But again, this is irrelevant for this situation, because going full twice has no extra value. So I think you should more carefully state your question.

Craig