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Best of Alan Krigman

Gaming Guru

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You Can't Rely on Averages when You're Dealing with Small Chances

9 December 2001

When casino outcomes have high or moderate possibilities of occurrence, it doesn't take a Methuselah's age of action before actual frequencies home in on statistically expected averages. As an example, there's around 21.5 percent chance of ending with a high pair at jacks-or-better video poker, so in 1,000 tries you'd expect 215 of 'em and would almost surely get between 200 and 230. A probability of a natural in multi-deck blackjack is about 4.75 percent. In 10,000 hands, this averages out to 475 - and you'd likely see from 450 to 500 of the li'l devils.

The situation is different for rare events, those with extremely low probabilities. Averages then become fickle forecasts. Progressive jackpots at Caribbean Stud offer good illustrations.

The probability of a royal in Caribbean Stud is one out of every 649,740 hands. After this many hands without a grand slam, the jackpot would be around $160,000. But, it's common for jackpots to reach $250,000 or $300,000, indicating that nobody's scored in over two or three times 649,740 tries. It's also not unusual to notice totals under $20,000 or $25,000 on several successive visits to the same joint, suggesting that the jackpot's been hit and reset more than once during a relatively short interval.

The following table shows the effect. The percentages are the probabilities of from zero to five and over five royals after enough hands for the expected number to be one, two, or three.

Theoretically
expected number of royals
1 2 3

Actual
no of
royals
Number of hands
649,740
1,299,480
1,949,220
0
36.8%
13.5%
5.0%
1
36.8%
27.1%
14.9%
2
18.4%
27.1%
22.4%
3
6.1%
18.0%
22.4%
4
1.5%
9.0%
16.8%
5
0.3%
3.6%
10.1%
over 5
0.1%
1.7%
8.4%

These figures afford several insights about low-probability events. The most important is that averages or expected values are not necessarily good indicators of real performance when a single random event can have a large relative impact. If you're dealing with the frequency of high pairs at video poker, a few more or less than the theoretical number don't have much significance after 1,000 rounds. Getting 210 or 220 - five less or more than the numero-noodniks predict - is no big deal.

But, look what happens at Caribbean Stud. Players should average one jackpot in 649,740 hands. But the table shows they indeed have the same prospect, 36.8 percent, of getting none as one. And the chance is pretty good, 18.4 percent, of getting two. Similarly, in 1,299,480 hands, two jackpots are expected; but one and two have the identical probabilities of occurring, 27.1 percent. And, there's a 13.5 percent chance of not getting any.

More, in a situation like that for blackjacks, you'd have cause for concern were you to get only half the expected naturals in 10,000 hands. The table shows, though, that while three royals are expected in 1,949,220 Caribbean Stud hands, players are as apt to get only two - 22.4 percent, and wouldn't ask the Supreme Court to order a recount were they to get only one, or none at all, the chances there being 14.9 and 5.0 percent respectively.

On the flip side of the coin, the casinos might be quick to call in their fraud squads if they dished out more than five jackpots within about 2 million hands. However, this can be anticipated by chance rather than hanky panky in 8.4 percent of all 2-million hand cycles. Here's another way the bosses could view the same figure. They can be 100 minus 8.4 or 91.6 percent confident they won't have to pay jackpots to more than five solid citizens in 2 million honest hands. Enough not to stew about sacrificing their silk-suit salaries; not so much to fold the game if it happens.

There's also the issue that the statistics of high and moderate probability events may bear on individuals. But the arithmetic of remote phenomena applies to whole populations - no one member of which gets near the mean value. This imbalance was implied by the improvisator, Sumner A Ingmark, in his impressive amoebaeum:

Why are averages unfair?
Take a million paupers and a trillionaire.
With what should I them compare?
To a million people with an equal share.

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.