ANSWER - The way it should be done
Or you could just click in a cell and type =combin(10,2). Done.
The calculation of the exact probability of flopping only a 4-flush draw (not a flush, not a SF, not a straight, not a straight draw), when holding JTs, is extremely tedious! (Pairing your hand or a pair on the flop are acceptable.)
It need not be extremely tedious if you know what the frick you’re doing, and choose an appropriate partition for the sample space. I claim you are not a proper judge of what must be tedious and what is not tedious, as I will now prove:
First of all, since we are allowing pairs, I’m assuming you want to allow 2-pair and 3-of-a-kind as well. If you want to disallow any of these, including pairs, let me know and I will do that just as easily, I would just have to change a few numbers in what follows. The probability we are after is equivalent to the probability of flopping exactly a 4-flush draw without the cards for a straight draw, since if we don’t flop the cards for a straight draw, then we don’t flop a straight either. We can always flop a 4-flush draw without the straight draw cards if we flop a 4-flush draw with no more than 1 of the 4 denominations 8,9,T, or K (more than one of the same denomination is OK, like 888 since we are allowing this). We can also always do it when we flop only the denominations 8,K. We can also always do it if we flop only 8,Q or 9,K so long as we don’t flop exactly 8,Q,A or 7,9,K, since these are the only double belly buster draws. These are all the cases, since the other combinations of two cards 8,9; 9,Q; and QK are straight draws. So now we count the cases.
No 8,9,Q,K: C(7,2)*27 = 567
Exactly 1 denomination of 8,9,Q,K: 4*7*30 + C(7,2)*12 = 1092
8,K: 2*3*7 + 1*33 = 75
8,Q but not 8,Q,A: 2*3*6 + 1*30 = 66
9,K but not 7,9,K: same as above = 66
Where I have added two terms, the first is for exactly 1 of the listed cards being a flush card. The second term is for either exactly 0 or 2 of the listed cards being flush cards, as appropriate to the case. Exactly 1 denominaton means multiple of same denomination still OK.
(567 + 1092 + 75 + 66 + 66)/C(50,3) = <font color="red">9.5%</font color>. This is the probability of a clean flush draw, exact to within the probability of a blunder, which in the limit should go to zero. [img]/forums/images/icons/laugh.gif[/img]
Now the probability of a flush draw including the times we also make a straight draw or straight is
C(11,2)*39/C(50,3) = 10.9%. So the probability of making a flush draw while at the same time also making either a straight draw or straight is 10.9% - 9.5% = 1.4%. The probability of making a straight draw or straight including the times we also make a flush draw is [ 3(12*42 + 4*33 + 6*4*2) + 4*4*4*2 ]/C(50,3) = 11.1%. Therefore, the probability of a straight draw or a straight is 11.1% - 1.4% = 9.7%. The probability of a straight including straight-flush is 4*4*4*4/C(50,3) = 1.3%, so the probability of a clean straight draw is 9.7% - 1.3% = <font color="red">8.4%</font color>. The probability of a straight draw or a flush draw or both is 9.5% + 11.1% - 1.3% = <font color="red">19.3%</font color>.
This is in agreement with the figures given earlier for the clean straight draw, but not for the clean flush draw. Since we used one to get the other, the earlier figures do not appear to be consistent with each other.
So in summary:
P(clean flush draw) = 9.5%
P(clean straight draw) = 8.4%
P(clean straight draw OR clean flush draw not both) = 17.9%
P(clean straight draw OR clean flush draw OR both) = 19.3%
P(both clean straight AND clean flush draw) = 1.4%
Now was that so tedious?
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