Back to Back Royals, Other Odds

[Mike]

Here's John's third article:

There’s no telling when your number is going to come up, but when it does, there’s nothing to prevent the random number generator from calling it again posthaste.

A reader once phoned with an experience to make video poker players everywhere jealous. It seems she was playing 25-cent Double Bonus Poker on II when she hit a royal flush, worth $1,000 for five quarters played.

After tipping the change person who helped with the payoff, the player decided to stay and play for a while longer. And a couple of hours later, that change person was startled to see a royal flush on the screen of a 25-cent progressive machine.

“Oh, it’s YOU!” she exclaimed.

Our player had hit a second royal, this time worth a progressive jackpot of $1,070.

It doesn’t stop there. After the second jackpot had been paid, the player hit the “Maximum Bet” button one more time. Casinos request that jackpot winners play one more hand on video poker or one more spin on slots so the winning combination doesn’t remain on the screen or reels. They fear that players who see the jackpot has just been hit will avoid the machine. This extra hand brought an initial deal that included four parts of a royal flush.

Alas, there were not to be consecutive royals here. The one-card draw brought no payoff.

"Wouldn’t that have been something?” the player asked. “The odds on that must be out of sight.”

She knows a little something about the chances of drawing a royal flush. On an earlier casino trip, she hit a royal, but lost back the entire $1,000. Chalk that up to experience, poor money management and a house edge that catches up to all gamblers eventually.

On each royal, she overcame odds of about 48,047 to 1 --- expert strategy on full-pay Double Bonus Poker brings royals an average of about once per 48,048 hands. The odds of hitting two in a row are 1 in 48,048 multiplied by 48,048, or about 1 in 2.31 billion. The chances that this player would hit two in the space of a few hours depends on how many hands she played. If she played continuously for four hours at a pretty good speed, she might play 3,000 hands. That being the case, her chances of hitting two royals in that time are about 1 in 770,000.

Those are the odds as the player walks up to the machine to begin play, and those remain the odds as play continues. If 1,000 hands pass with no royal flashing across the screen, the odds of hitting the jackpot on the next hand remain 1 in 48,048.

But the chances of hitting two royals change the instant the first one is hit. Given that the first royal already is in the bank - or purse, or pocket - the chances of hitting a second at Double Bonus Poker now are 1 in 40,048, the same odds as hitting one royal at any other time.

Look what happens to the odds against royals on consecutive hands. Before the first hit, our player has one chance in 1.6 billion to hit royal flushes two hands in a row. After the first royal, there’s one chance in 48,084 that another will come up on the next hand.

And when four cards to a royal are dealt to begin the next hand, the odds go into free-fall. The player now has a 1-in-47 shot of completing that second consecutive royal flush. Why? Cards in video poker --- except in Joker games, which use 53 cards --- are dealt from a randomly shuffled 52-card electronic deck. The player already has seen five cards, including four parts of a royal. Forty-seven cards remain in the deck. One of those 47 will complete the royal if it is dealt to replace the one card discarded by the player.

Note that the odds against hitting a royal never really change. It’s our conditions that are changing here. We start by talking about the odds against hitting two royal flushes. Once the first has been hit, we’re really only talking about the odds against hitting one more royal. And once we know what first five cards are, we have yet another set of odds, because we’re dealing with many fewer possibilities.

With the exception of blackjack, the principle is the same in almost any casino game. The odds against any given event occurring on this roll or spin as they are on the next, or on a trial next week.

If a slot machine’s random number generator is programmed to yield the top jackpot about once in 10,000 pulls, there is a 1 in 10,000 chance of hitting on the next pull. Even if the machine has gone 9,999 pulls in a row without paying the jackpot, there is no better chance of hitting on the next pull. And even if the jackpot was hit on the last pull, the chance of hitting on the next is no worse. Either way, it’s still 1 in 10,000.

At the craps table, it’s said the dice have no memory. Before the dice are rolled, there’s one chance in 36 that a 12 will come up. If a 12 is rolled there’s once chance in 36 that the next roll also will be a 12. The result of the last trial has no effect on the next.

That’s a simple reality that worked to our video poker player’s advantage this time. She overcame the odds and hit a royal. And later, her odds of hitting again were no worse than hitting the first: The machine had no way of knowing she already was a winner.

She knew what to do with her winnings this time, too. She walked away richer by almost the entire $2,070.

- John Grochowski

Thanks for reading the column, I hope you enjoyed. If you'd like to leave comments or questions for John, make sure to register for free so you can post replies (and disable advertising).