Intelligent Design

Saturday Fun: When the Lottery Bet Has a Positive Expected Value

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This is one of those very rare times when the lottery bet has a positive mathematical expected value. Expected value is calculated as: (Amount possibly won * probability of winning) minus (Amount of bet * probability of losing).

The probability of winning Mega Millions is 1 in 302,575,350. The next jackpot is $904 million (cash value of $1.6 billion annuity). The expected value is ($904,000,000 * 1/302,575,350) minus ($2.00 * .9999999999999999999) = $0.98.

This means on average in the long run, for every $2.00 ticket you buy, you would expect to win $2.98 if the jackpot were always $904 million.  Of course, you still lose the whole $2.00 every time you lose, which is almost always.  Still, on average, over the long run, the expected value is positive ($2.98 – $2.00 = $0.98).

In the long run, it is a good bet. Of course, the problem is there is no long run. You only have a single shot at it. To achieve the long run average expectation, you would have to play several hundred million times.

For this single jackpot, like all other jackpots, you are mathematically almost certain to lose your entire $2.00. Still, at least this time, the bet is not a mathematical loser from the git-go like all those other times you played. Interestingly, if you could afford to buy several hundred million tickets, the math says you should. 

Someone said the lottery is a tax on those who are bad at math, and that is almost always true.  This is one of those rare instances where it is not. The math says play. This time the lottery is a tax on those who cannot afford to play several hundred million times.

This is a special case of the phenomenon known as “gambler’s ruin.” Gambler’s ruin is one of the reasons casinos are so profitable. Gambler’s ruin says that in the long run anyone with a limited bankroll will always lose to someone with a practically unlimited bankroll even if they have a mathematical edge (and especially if they do not). This is so because the unlimited bankroll can ride out bad short- to medium-term variance from the long run expected value, while that same variance would “ruin” the limited bankroll.

4 Replies to “Saturday Fun: When the Lottery Bet Has a Positive Expected Value

  1. 1
    Orloog says:

    You should account for split prices: the lottery tickets are not unique.

  2. 2
    Mung says:

    And don’t forget to factor in the probability of losing the winning ticket!

  3. 3
    jdk says:

    Very good post, Barry! 🙂

  4. 4
    Bob O'H says:

    Someone said the lottery is a tax on those who are bad at math, and that is almost always true.

    I’m not a big fan of that argument – it assumes (a) the only reason to play a lottery is for profit, and (b) there is a linear relationship between money and quality of life. I don’t think either is true: the latter certainly isn’t: a $900m prize will make a huge difference to my life, whereas losing $2 (or even $200) will make little difference. The former ignores why people do things: buying a ticket and anticipating the numbers coming up can be its own reward. We do a lot of things that don’t bring us monetary profit (watch TV, go to the opera etc.), and playing the lottery can be little different in this regard.

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