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1 - (1 - 15/392)^N.
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Could you please explain the thinking behind this formula?? I understand that (1 - 15/392) is calculating the probability that one face down hand doesn't have the flush, but I forget why we have to approach the problem from this angle.
Cheers
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If you take the probability that 1 hand does not have a flush and multiply it by itself N times (raise it to the Nth power), this gives you approximately the probability that N hands do not have a flush. This would be exact if the hands were independent, but actually they are not since if one hand has a flush, it changes slightly the probability that the other hands have a flush.
For the exact answer, you need the inclusion-exclusion principle (http://archiveserver.twoplustwo.com/showthreaded.php?Cat=&Board=&Number=417383&page=&v iew=&sb=5&o=&fpart=). The last example in that post has a similar calculation. If you only compute 2 terms of inclusion-exclusion, the approximation is better than the independence approximation. The data below shows a comparison of these two approximations for this problem. The first column is the number of hands N. The second column gives the exact probability that at least 1 hand has a flush to 6 decimal places computed by the inclusion-exclusion principle. The second column gives the result of the first 2 terms of inclusion-exclusion which are N*15/396 - C(N,2)*C(10,4)/C(49,4). The last column gives the result of the independence approximation 1 - (1 - 15/396)^N. As you can see, the error gets worse as the number of hands increases, but it increases much more slowly for the inclusion-exclusion approximation than for the independence approximation.
<font class="small">Code:</font><hr /><pre>
# hands P(flush) exact P(flush) 2 terms P(flush) independence
1 3.826531% 3.826531% 3.826531%
2 7.553947% 7.553947% 7.506638%
3 11.183750% 11.182248% 11.045925%
4 14.717432% 14.711435% 14.449780%
5 18.156475% 18.141507% 17.723385%
6 21.502350% 21.472465% 20.871725%
7 24.756520% 24.704308% 23.899593%
8 27.920436% 27.837037% 26.811598%
9 30.995541% 30.870651% 29.612175%
10 33.983266% 33.805150% 32.305586%
11 36.885034% 36.640535% 34.895934%
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