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Best of Alan Krigman

Gaming Guru

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Probabilities, Not Casino Bosses, Control the Poker Machines

28 May 1996

I talked to a sweet old lady in a casino lounge last week.

She: I don't play the poker machines any more.
Me: Why not?
She: They always give you cards you don't want.
Me: Which 'they' do you mean?
She: The casino bosses who control these things.

Trying to set the record straight, I began to explain the pseudo-random number generator that determines the cards a player gets after pressing the button. The glazed look I know so well clouded her eyes. So I tried another tack and suggested she think of the machine as an automatic way to shuffle and deal from a real deck.

"Then why do I always get something like a four when I need an ace?" she demanded.


"You got five cards to start," I answered. "Say you tossed one and need a six to make an inside straight. With no jokers, in a machine or a real deck, 47 cards remain. Only four are sixes. You take the next card. Chances of its being a six are four out of 47. Chances of something else are 43 out of 47. Instead, what if you need the six of hearts for an inside straight flush? Now only one card out of the 47 will do. Your chances drop to 1 out of 47. Chances of anything else are 46 out of 47."

"Like I said," she smiled, "it's loaded so you don't get the card you need. I'll tell you a secret. They don't want you to win."

"Can I quote you on that?" I asked.

Thinking about this chat later, I realized that good poker players know the likelihood of drawing to make various hands. Maybe not all the exact percentages - the permutations can get complicated - but close enough to decide whether to bet or fold.

Solid citizens who play the machines, though, can learn the optimum hold/discard rules without considering the actual probabilities in any situation. So, when they start with two pair and make a full house, they don't realize they've beaten odds of nearly 11-to-1. And, when they don't get just the right cards for a big payout, they think the machines are set to bedevil them.


Say you're at a Jacks-or-better draw poker machine. Or, you're at your kitchen table playing five-card draw. Either way, fate blesses you by dealing 5-D 10-H J-H Q?H K-H for starters. Naturally, you dump the 5-D. In the video version, the machine replaces it with one of the 47 cards you haven't seen - those still in the deck. In the kitchen contest, the dealer replaces the 5-D with one of the 47 cards you haven't seen - those the other players hold or have discarded and those left in the deck. The accompanying table shows what can happen in either case.

Although a big jackpot appears to be only one card away, you can see from the table it's not that simple. The difficulty comes from the probabilities associated with each final result. And these aren't arbitrary figures casino management loaded into the machine. They're inherent to five-card draw - video and kitchen games alike. They're the number of cards that can give you each result, divided by the 47 unknowns from which you draw.

Picture the probabilities by imagining a thousand video poker players lucky enough to start with the proposed hand. Expectations after the dust settles are 21 royals, 21 straight flushes, 150 flushes, 128 straights, 191 high pairs, and 489 firm convictions the machine was fixed to give them a card they didn't want.

Valiant video versifier, Sumner A Ingmark, voiced it vibrantly:


Cards left in the deck are what govern your chances,
No secretive plan vict'ry halts or advances.

TABLE
Probabilities associated with drawing one card
to a starting hand of 10-H J-H Q-H K-H
Card Drawn
Result
Number of
possible
cards
Probability
A-H
royal
1
1/47 = 2.1%
9-H
straight flush
1
1/47 = 2.1%
2-H 3-H 4-H
flush
7
7/47 = 14.9%
5-H 6-H 7-H
8-H
A-C A-D A-S
straight
6
6/47 = 12.8%
9-C 9-D 9-S
J-C J-D J-S
Q-C Q-D Q-S
high pair
9
9/47 = 19.2%
K-C K-D K-S
anything else
loser
23
23/47 = 48.9%
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.