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The gambler's fallacy is the mistaken notion that the odds for something with a fixed probability increase or decrease depending upon recent occurrences.
For example, in California we have a state run gambling operation called Superlotto. The idea is to pick 6 numbers and match them to six selected from 51 numbers. Sounds easy. The odds of doing so? Here is what happens in a typical week. On July 25, 1998 the numbers were: 5, 7, 21, 32, 44, 46. The Jackpot was $16,000,000. There were no tickets with all six numbers. 170 tickets matched 5 numbers and won $1,588 each; 9,715 matched 4 of 6 numbers for $72 each and 176,657 matched 3 of 6 numbers for $5 each.
If you programmed a computer to randomly generate six different numbers every second taken from the numbers 1 through 51, you would have to wait nearly seven months before every combination came up at least once. (Assuming a combination can't come up again until every other combination has come up.)
The odds of matching 6 of 6 numbers are 1 in 18,009,460; 5 of 6 are 1 in 66,702; 4 of 6 are 1 in 1,213; 3 of six are 1 in 63.
The odds of winning anything are 1 in 60.
If you buy 100 tickets a week, you can expect to win the jackpot on average every 3,463 years. If you buy $25,000 worth of tickets a week, you can expect on average to win about every 14 years. If you expect to live 50 more years, you should buy $6,927 worth of tickets a week if you want to have a good chance of winning the jackpot in this lifetime. Of course, if you do, you may not even break even. You could well be about $2,000,000 in the hole, depending on when you win.
However, if you would be satisfied with getting 5 out of 6, you will have a much easier go of it. You are likely to get 5 out 6 every 12.8 years on average if you buy 100 tickets a week. However, you will have spent nearly $67,000 to win about $1,500.
If you want to "guarantee" yourself to be a "winner," buy about $120 worth of tickets a week. On average, you are likely to take home, before taxes, about $10 a week. Thus, to be a "guaranteed winner" you need only lose about $110 a week. What could be easier? (This "guarantee" comes with a limited warranty of no value and is based upon payouts for the week of July 25, 1998.) Of course, your winning at all is purely theoretical. You may never win regardless of how often you play and how much you spend. Mathematical odds are based on theoretical chance, which is not the same as real odds in the real world. Theoretically, there is a one in two chance of a coin flip coming up heads, but in reality heads might come up more or less than five times in ten flips.
You might think that you can beat the odds by either selecting numbers that have not been chosen in recent drawings, or by selecting numbers that have come up more frequently than expected in recent drawings. In either case, you are committing the gambler's fallacy. The odds are always the same, no matter what numbers have been selected in the past. This fallacy is commonly committed by gamblers who, for instance, bet on red at roulette when black has come up three times in a row. The odds of black coming up next are the same regardless of what colors have come up in previous turns. Dice players frequently commit this fallacy. When they see a player make his point several times in a row, they think the odds of him making his point again diminish. They don't. Those odds are fixed and they never change. The odds of making any given point differ, of course, depending on what the point is. For example, there is a six in thirty-six chance that a 7 will be rolled, but only a one in thirty-six chance that a 2 or a 12 will be rolled.) I won't comment on the futility of superstitious acts like blowing on the dice or talking to them in an attempt to influence the outcome except to say that such rituals are based on magical thinking and are commonplace in our species.
Lotteries seem to share something in common with illegal pyramid scams: in order for anyone to win big, many people must lose just about everything they bet.
(Note: None of the above math has any monetary value and has not been verified by our statistician. Use at your own risk.)