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The Random Texan

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Dilletante * Raconteur

There’s fair, and then there’s fair…

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by

therandomtexan

in

We had a surprising discussion last week in one of my probability
classes. Zach remarked that he had tried in vain to
convince a friend that the Texas Lottery was not fair, since there is a
"house advantage." The friend was unconvinced, arguing that the lottery
must be fair, since everyone has the same chance of winning. Spotting a teachable moment*,
I steered the discussion to the different concepts of fair. Friend of
the wizard thinks it’s "all men are treated equal," while modern
probabilists think it’s "the expected payout is the price of the bet."
Some folks are still stuck in the 16th century, which is why states and
casinos can profit at numbers running.

*I hate this phrase. Once students have some tough problems they want to solve, every moment becomes teachable

2 responses to “There’s fair, and then there’s fair…”

  1. Dawn Avatar
    Dawn

    For those of us trying to keep up, "The expected payout is the price of the bet" means that the more often I bet the better my chances are of winning? Help a blonde out, lol… Aube

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  2. Spunky Avatar
    Spunky

    Aube: Buying more tickets won’t change the expectation, or make the game "more fair." Consider a simple raffle, with 100 one-dollar tickets. For a fair bet, you expect to win $100 with a chance of 1/100 costing $1 ($100 x 1/100 = $1). If you want to increase your chances, you can buy more tickets; let’s say you bought 12 tickets. That gives 12 chances out of 100, and $100 x 12/100 = $12, which is what you paid for the tickets. In both cases, payout times probability equals price, so the game is fair. The more common case is a fund-raising raffle, 100 one-dollar tickets with only a $50 prize. Now the expected value for a ticket is $50 x 1/100 = 50 cents, only half the price. This ain’t fair. Same for 12 tickets, since $50 x 12/100 = $6. You can still increase your chances, but you can’t make an unfair game fair. The critical distinction in the definition of a "fair bet" is that the financial risk be balanced between the gamblers and the house, not just among the gamblers. The house has a low risk of paying, so they get a small reward. Each gambler has a high risk of losing, so each expects a high reward. What surprises me is that so many people play when they know the house has rigged the odds.

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