Hitting the club royal

I was at Bally's again last night, had another profitable and reasonably enjoyable session. I might actually come to like the place, in spite of the noise factor.

The jackpot for hitting a royal flush in clubs, using both hole cards, is almost $12,000 now. Several times last night I had two of the five required cards in my starting hand, which makes one's heart beat just a little bit faster when the flop is about to come.

It got me wondering about the probability of hitting a royal flush, given two qualifying down cards. I just worked it out, using the assumption that I'll see a full board of five community cards every time:

Suppose I start with the Ac-Kc. I obviously have to get a board that includes exactly the Qc, Jc, and 10c. I don't care what order they come in, and I don't care what other two cards accompany them. After specifying the five cards that matter, there are 47 cards left in the deck, from which any two can come. There are 1081 different ways to draw two cards from 47, so there are 1081 different boards that win me the jackpot.

But how many total possible five-card boards are there? Well, I have two cards in my hand, so those are not available to come. You can calculate this in Excel by entering into any cell the formula =COMBIN(50,5). Doing so tells us that there are exactly 2,118,760 different ways to draw five cards from the 50 that are left in the deck.

The final step is to compare the 1081 boards that win to the universe of 2,118,760 possible boards. The answer is 0.00051, or about 0.051% of the time. It will happen only once out of every 1960 times that I have a qualifying starting hand.

Looks like it's going to take a lot of hours sitting in the noise.