Sucker bet

From Hobson's Choice

In casino gambling, a sucker bet is a wager offered by the house that provides a sure flow of revenue. The odds of winning, times the payout, are much less than the cost of playing. An obvious example of a sucker bet is a lottery:

That the chance of gain is naturally over-valued, we may learn from the universal success of lotteries. The world neither ever saw, nor ever will see, a perfectly fair lottery; or one in which the whole gain compensated the whole loss; because the undertaker could make nothing by it. In the state lotteries the tickets are really not worth the price which is paid by the original subscribers, and yet commonly sell in the market for twenty, thirty, and sometimes forty per cent. advance. The vain hope of gaining some of the great prizes is the sole cause of this demand. The soberest people scarce look upon it as a folly to pay a small sum for the chance of gaining ten or twenty thousand pounds; though they know that even that small sum is perhaps twenty or thirty per cent. more than the chance is worth.
Adam Smith, Wealth of Nations, I.x.30

In modern lotteries, the payout may be nominally about the same as the receipts from ticket sales, since it is awarded over a long period of time, allowing the lottery to collect interest.^[1]

The term "sucker bet" is often used to refer to certain categories of card game in which the player is allowed to win an inconsequential amount initially, and then drawn into making much longer bets for higher stakes. Often it is used to refer to any bet in which the person making the bet is extremely misinformed about the true odds.^[2] A popular example is the "birthday bet," in which a party of 25 are in a room. Someone offers to bet 5:1^[3] that two people in the room have the same birthday. The implication is that there's a 16.7% chance that there is a pair of coincidental birthdays. Someone else thinks about this bet for a moment. There are 365 calendar days, and only 25 people. The odds that any one person has the same birthday as any of the others is, indeed, (1-(1/365))²⁵, or 6.62%. Betting in the opposite direction, viz., that there are no coincidental birthdays in the group, implies 1/5 odds, or 83.3% probability of winning.

This is a sucker bet, however, because the tentative estimate of 6.62% probability of a coincidental birthday is true for each individual in the group. For all the individuals in the group, the correct figure is 56.1%! In other words, it's more likely than not that there is, indeed, a coincidental birthday in the group!^[4]

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Notes

↑ Assuming a 7% time discount, a payout of €100,000 over 20 annual payments is worth only €56,678; if over 10 such payments, it's worth €63,893 (this is assuming the first payment occurred immediately after winning).
↑ Gayle Mitchell, Easy Casino Gambling: Winning Strategies for the Beginner, Skyhorse Publishing (2007) definition on p.130
↑ 5:1 odds means that the person betting on the thing happening will lose n euro if the thing doesn't happen, and win 5n if it does. The first number is your winnings as a multiple of the stake (second number). Taking another example: supposing you bet on something at 8/15 odds. Your stake is €100. For every €15 that you stake, you win €8. Divide €100/15 to get 6.66, then multiply that by 8 to get €53.33. When you collect your winnings, you'll receive €153.33 (i.e., your stake plus winnings). Odds of 8/15 imply that a probability of 15 ÷ (8+15), or 65%.
↑ A group of 25 people includes 300 dyads, or pairwise relationships. The odds of a coincidental birthday are thus 1-(1-(1/365))³⁰⁰, or 56.1%.

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