I have had to analyze some multi-deck poker games recently. Suppose I am dealt 7 cards from an 8-deck “shoe”, and I have to make the best 5-card hand out of 7.

What is the probability my best hand is a flush (meaning no 5 of a kind, 4 of a kind, full house, straight flush, or royal flush are also obtainable)?

What is the probability my best hand is a straight (meaning no flush is attainable either)?

I solved this in a spreadsheet, but it was at the borderline, if it had been any more complicated I would have written a program.

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Here’s a hint. Calculate the probability of a straight first, by treating separately the different denomination distributions compatible with a straight (7 different denominations, 1 pair, 2 pairs, 3 of a kind). For each of those distributions, count how many include a straight, calculate how many ways to assign the suits, and calculate the probability you have a flush (which doesn’t count as a straight) by looking at all the different ways 5, 6, or 7 cards could be the same suit (for example, when you have 2 pairs, and a flush, you could have all 7 cards be the flush suit, or all but one of the odd cards, or all but one of the cards from one of the pairs, or all but two of the odd cards, or all but one odd card and one from a pair, or all but one card from each pair, or all but the two cards from one of the pairs).

The overall probability of a flush is now easy because you’ve done most of the work already, the only additional denomination distribution you need to look at is 3 pairs (because other distributions will give you full house, 4 of a kind, or 5 of a kind and so won’t count as flushes). Then at the end you have to count straight flushes and exclude them.

I told you it was hard. This took me about 2 hours to do in Microsoft Excel, including checking everything for correctness. But 2 hours was still less time than it would have taken me to write and debug a program even if I don’t count all the time the program took to run through the 7-card combinations.