How many straight flushes are possible with wildcards?

Answer

In a normal deck there are 40 possible straight flushes: 10 in each suit. With two wildcards this number jumps to 624. This number is not easy to derive.

To calculate we count the number of combinations in three situations:

• when the hand has no wildcards
• when the hand has one wildcard
• when the hand has two wildcards

These three results can then be added together to get the total count. This can be done since they are exclusive sets: there is no hand which exists in multiple categories.

The Naive Approach

The answer to the first item is already known: there are 40 possible straight flushes without wildcards. It seems like it should be easy to calculate the second one group. We simple calculate how many ways one card in the hand can be replaced with a wildcard. Since there are five cards, that means there are 5 ways in which one can be replaced with a wildcard. That gives us 40 * 5 = 200 possible straight flushes with one wildcard. The game has two wildcards though, so we double that number to 400.

The third possibility is calculated similarly. Instead of replacing one card we want to replace two cards with a wildcard. This is a simple combination calculation; there are (5c2) = 10 ways to pick two cards from five. That results in 40 * 10 = 400 additional straight flushes.

All together that gives us 40 + 400 + 400 = 840 total combinations. This gives us perhaps a rough estimate, but it is quite wrong.

Some Correction

The count of 840 is too high. Consider the hand 8♥ 7♥ 6♥ 5♥ 4♥ and replace the lowest card with a wildcard to get 8♥ 7♥ 6♥ 5♥ RJoker. Now consider the hand 9♥ 8♥ 7♥ 6♥ 5♥ and replace the highest rank with a wilfcard to get RJoker 8♥ 7♥ 6♥ 5♥. Those are the same cards we had in the first case!

Here it is true that the wildcard can be used to create a nine-high or an eight-high straight. So technically it is two different straight flushes, but it is only one combination of cards. Since we usually care about probabilities we need to consider this as only a single combination. The naive calculation ignores this and ends up counting the same combination multiple times. Correcting this is not simple.

Logically we can say that in the one wildcard case we simply don't replace the lowest most card. In that way we don't have to worry about a low straight becoming a high straight. We can't forget about the highest straight however, A♥ K♥ Q♥ J♥ T♥. In that case we can replace the lowest card since there won't be a higher straight. So for 9 of the straights we have 4 cards to replace, and the high straight we still have 5 cards to replace. With four suits and two wildcards that results in (9*4 + 1*5) * 4 * 2 = 328 five of a kinds using one wildcard. A bit lower than the 400 we calculated previously.

For the two card case we can take the same approach and anchor the lowest card to prevent overlaps, taking care that on the highest straight we allow it again. So again in 9 cases we have 4 cards to choose from and in 1 case 5 cards. Each time we choose two of the cards. For all 4 suits that gives us (9*(4c2) + 1*(5c2)) * 4 = 256 straight flushes using both wildcards.

In total we get 40 + 328 + 256 = 624 possible straight flushes if two wildcards are used. This is only slightly lower than our naive estimate of 640 such combinations. Such calculations are quite tricky and easy to get wrong.