# Probability

The answer is 1 in 5857.

This is the trickiest exercise so far. When we add the two jokers to the deck, the total number of combinations goes up, because you now have a 54 card deck instead of a 52 card deck. Instead of 52C5, we now have 54C5, which equals 3162510 instead of 2598960. The number of possible straight flushes goes up as well. For every "natural" straight flush, there are now a nunber of "wild" straight flushes. Adding 1 joker to the deck gives us 5 possible variations on each natural straight flush, since the joker could replace any of the 5 natural cards. However, in practice, no one would ever replace the low card with a joker because that would produce a lower-scoring straight flush. For example, if you had 5,6,7,8 and 1 joker, you would use the joker as 9, not 4. Therefore, even though there are 5 possible variations on a wild straight flush with 1 joker, 1 of them can be discarded so there are really only 4.

Tricky enough for you? Now, if we add two jokers, we have 2 wildcards in the hand that could be arranged in 5C2, or 10 ways. But once again, some of those orderings would never actually be used in the game of poker, because you would never use the jokers as the bottom cards in a straight flush. Specifically, you would never use JJ678, J5J78, J56J8, or J567J. That means you can discard 4 of the 10 combinations, leaving 6. This, in addition to the 2 possible variations of 1 wildcard (with 4 potential orders for each), means you have a total of 6+4+4=14 variations of wild straight flush for each straight flush.

Putting it all together, we have 36 straight flushes, plus 14*36=504 wild straight flushes, for a total of 540. This means the odds of drawing a straight flush from a deck of 52 cards plus two jokers is 3162510 divided by 540, or 1 in 5857.

Straight Flush with Jokers #1

Just for fun, feel free to click on the cards to see all 540 possible combinations, if you're patient enough to wait that long. By the way, contrary to popular belief, this kind of problem does not necessarily become easier if you write a computer program to brute-force the calculation rather than doing it the elegant way. That's because the biggest potential pitfall is to forget a rule you need, and you can just as easily forget a rule when writing a program as you can when solving it analytically. Worse yet, when you write a program your attention is divided between mathematics and the vagaries of computer language.

PS. And in case you're wondering, yes. Your eyes aren't playing tricks on you, I actually used a picture of Jack Nicholson's Joker from Tim Burton's "Batman" movie.