h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

The statistics presented in this article are misleading. Playing all your tickets at once does indeed increase your odds of winning the lottery, but not nearly as dramatically as the article appears to suggest, and it does not alter your expected earnings. What it does do is alter the variance of your expected winnings - the probability distribution of your expected winnings is tighter if you play multiple tickets at once. Sound counter-intuitive (or just too complicated)? Time for a heavily simplified hypothetical example then...

Imagine I am tossing a coin and allowing you to bet on it, giving you £10 for guessing correctly whether it lands on heads or tails. You are going to make two bets. You can either bet twice at once, with one bet on heads and one bet on tails (the equivalent of buying a ticket for each of the 13,983,816 combinations in a lottery draw) or you can place one bet only, then place another single bet on a second coin toss (the equivalent of buying lottery tickets for successive draws).

If you make two opposing bets at once, on the same coin toss, you are certain to win the £10 prize.

If you make the two bets on different coin tosses, there are four outcomes with equal probabilities (1 in 4):
1) You win both games, making £20
2) You win the first game but lose the second, making £10
3) You lose the first but win the second, making £10
4) You lose both and get nothing.

By adopting the the first tactic, the two bets at once, you are certain to win: your overall chance of winning is 1, but you miss out on the possibility of winning twice. Your winnings are £10.
By adopting the second tactic, you only have a 3/4 chance of winning, as 1/4 of the time, choosing this tactic will result in two straight losses. But, your projected winnings, calculated on the basis of the probabilities of the different outcomes, are are still £10, calculated as follows: (1/4 * £20) + (1/4 * £10) + (1/4 * £10) + (1/4 * £0) = £10

So, chances of winning at least once change, but expected winnings do not. What changes is the probability distribution of your expected winnings. The lottery problem, while much bigger and more complex, conforms to this same principle.


If you're still not convinced because the example is too simple, then heres the maths for buying two lottery tickets on the same day vs buying two on different days. For simplicity's sake we'll assume the jackpot is £5m and is the only prize - the fact that it isn't doesn't alter the conclusion.

On the same day: probability of winning = 2 in 13,983,816 = 1 in 6,991,908 (about 1 in 7m as the article correctly stated)
Expected winnings: £5m x 1 in 6,991,908 = £0.715112384201852

On different days: probablilty of winning *only* on the first day (which we'll call prob1) = 1 in 13,983,816 x 13,983,815 in 13,983,816 (because you don't win on the second day)
Probability of winning *only* on the second day, prob2 = the same as prob1.
Probability of winning on both days = 1 in 13,983,816 x 1 in 13,983,816 (= about 1 in 200 trillion)
To get the probability of winning on at least one day, calculate p1+p2+p3, which equals 1 in 6,991,908.25. So you're only very very slightly less likely to win the lottery by playing on two subsequent days than playing twice on the same day. (This probability discrepancy gets larger and larger when considering a greater number of tickets.)

Expected winnings = (£5m x prob1) + (£5m x prob2) + (£10m x prob3) = £0.715112384201852, the same as for buying two tickets on one day.

The fact that you are ever so slightly less likely to win if you play on subsequent days, rather than multiple times in one day, is counteracted by the fact that you've got a tiiiiny chance of multiple wins, making your expected winnings (i.e. the average (mean) winnings between many people adopting these same tactics) the same, but with a higher variation in their probability distribution. As in the coin toss example.

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SO - the tactics described in the original article help in terms of risk management, but not in terms of expected winnings. To simplify, if you're the type of person who would rather win £100 exactly than have a 3/4 chance of winning £50 and a 1/4 chance of winning £200, then playing all your lines at once and never playing again is the better tactic for you. Although the best tactic is still just not playing the lottery

Correction: I should have said a 3/4 chance of winning £50 and a 1/4 chance of winning £250 at the end of my post...