Ask the Slot Expert: What are the odds of being dealt a royal flush?

Question: I recently was dealt a royal flush. What are the odds of being dealt a royal flush?

Answer: Congratulations. All royals are great, but dealt royals are special because they happen so rarely. I was lucky enough to be dealt a royal twice last year, an occurrence I doubt will ever happen again.

To find the odds of being dealt a royal, we have to find two numbers: the number of ways to make a royal and the number of ways to deal a five-card poker hand. Notice that I didn't say the number of ways to deal five cards. That phrase implies that the order of the cards is significant. Order is irrelevant in a poker hand -- at least on most machines -- so we need to find the number of five-card hands.

The first number is easy. There is one royal flush per suit and four suits in the deck, so there are four royal flushes.

For the second number, there are 52 ways to choose the first card in the hand, 51 for the second, 50 for the third, 49 for the fourth and 48 for the fifth. We multiply those numbers together to get 311,875,200. That's the number of ways to deal five cards when order is important. Mathematically this is called a permutation.

We don't care about the order of the cards in video poker. The mathematical term for this calculation is a combination. We have to divide by the number of ways to deal five cards.

How many ways are there to deal five cards? There are 5 cards for the first position, 4 for the second, 3 for the third, 2 for the fourth and 1 for the fifth. Multiplying these numbers together gives 120.

Dividing 311,875,200 by 120 gives 2,598,960. That's the number of five-card poker hands that can be dealt from a deck.

Odds are the ratio of the number of ways for an event to not happen to the number for the event to happen. Out of the 2,598,960 dealt hands, four are royal flushes, so there 2,598,956 way to not be dealt a royal flush. The odds are 2,598,956 to 4, or 649,739 to 1.

The probability of being dealt a royal is 4 divided by 2,598,960 or 0.00000154. That's a lot of zeros.

One other way to give the likelihood of an event is to give what Don Catlin called chances. I think this way of expressing the likelihood of an event is what most people mean when they ask for the odds and it's what most state lotteries refer to as odds. The chances of being dealt a royal flush are 4 out of 2,598,960, or 1 out of 649,740.

Speaking of odds, a few weeks ago a reader asked whether the odds of hitting the progressive were the same on Wheel of Fortune machines with different max bets. In my answer, I referred to old Nevada Regulation 14.045, which said that on machines connected to a common payoff, either the required bet had to be the same on all machines or the odds of winning the progressive had to be in proportion to the amount bet. The regulation specifically dealt with machines of different denominations participating in the same wide-area progressive, but the spirit of the regulation was that players betting more should have a better chance at the big prize.

That section of the regulation was amended about seven years ago and now just says that the machines "shall display the rules of play and the payoff schedule."

I found two Wheel of Fortune machines vying for the same progressive with different max bets. These machines have a plaque that says, "The odds of winning the Wheel of Fortune progerssive jackpot times the maximum wager is equivalent for all Wheel of Fortune progressive machines."

Let's express this text as a formula for two machines with max bets of $2 and $5, as in the original question. We don't know what the odds are, so let's just call them O2 and O5. The text says that the odds times the max bet are equivalent. The formula is 2xO2 = 5xO5. Doing a little division to solve for O2, we get O2 = 2.5xO5.

In other words, you're two-and-a-half times more likely to hit the progressive when you bet $2 on the two-coin machine than when you bet $5 on the five-coin machine.

Somehow I don't think that's what they mean -- and I hope that's not what's happening. You should be two-and-a-half times more likely to hit the jackpot when you bet $5.

Send your slot and video poker questions to John Robison, Slot Expert™, at slotexpert@slotexpert.com. Because of the volume of mail I receive, I regret that I can't reply to every question.

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