Topics Probability

Question My girlfriend and I had a discussion about probability as it relates to a weekly lottery draw. She argued that the probablity of winning remains the same from draw to draw, and because of this anyone who plays the lottery more than once stands no greater chance of winning than someone who only plays it on one occasion. Against this, I argued that because any lottery operates with a finite series of numbers, given enough draws all possible combinations will eventually have appeared at least once, and as such someone who plays more than once stands a greater chance of winning. I also claimed that the probability relating to each draw is different from that which relates to a succession of draws (again because of the finite series of numbers). Which of us is right?

Accepted:
September 28, 2006

Comments

Someone who plays the lottery more than once stands a greater chance of winning than someone who plays it on only one occasion. Compare: The chance of rolling "six" once with a fair die is greater if you roll the die twice than if you roll it once. The chance of your rolling "six" on one toss is (naturally!) 1/6. Your chance of rolling "six" once in two tosses is 1 minus your chance of rolling 1-5 on the first toss and 1-5 on the second toss, i.e., 1 - (5/6)(5/6), which equals 11/36 -- which is 5/36 more than 1/6.

However, you said that given enough trials of the lottery (having a finite number of tickets), every ticket will eventually win. That's not true. It's like saying that if you are tossing a fair coin, then it is guaranteed that a head will eventually appear. That's not true. You *could* get all tails. As the number of throws increases, the chance of getting all tails diminishes, and with an infinite number of throws, the chance of getting all tails is zero. But that does not mean that it is impossible to get all tails. It merely has a smaller likelihood than any finite non-zero number.

It might help to remember that any particular sequence of outcomes (whether it is TTTT... or HTHT...) is exactly as likely as any other (if the coin is fair). So even in an infinite sequence of tosses, it is possible to get all tails, just as any other sequence of outcomes is possible. It is just very unlikely.

There is a difficult problem lurking here. You might say: In the case of an infinite sequence of tosses, each sequence of outcomes has a likelihood of zero (I said above). So since the sum of their likelihoods is the likelihood that one of them occurs, and the sum of their likelihoods (0 + 0 + 0...) is zero, the likelihood that one of them occurs is zero. But that can't be! If a coin is tossed an infinite number of times, then it is certain (a probability of 1) that some outcome or other will occur (either HHH... or HTH... or whatever).

This is indeed paradoxical. Some philosophers would say that the principle of "countable additivity" is false. I used that principle above. It says that the probability that one of an infinite number of mutually exclusive outcomes occurs is the sum of the probabilities of the individual outcomes. The analogous principle clearly works for a finite number of outcomes. But as you can see, it isn't obviously correct for an infinite number of outcomes.

Marc is right that if you play the lottery more than once, your chance of winning at least once is higher than if you only play once. However, there is another possible interpretation of your question. Supppose you have played the lottery many times and lost every time. Is your chance of winning the next time higher than if you hadn't played before? Some people think that your chance of winning is now higher because the lottery "owes" you a win. But this is wrong; your chance of winning the next time is exactly the same as if you hadn't played before.

This does not contradict Marc's answer. This is because probabilities depend on the precise situation, and when the situation changes, the probabilities can change. Let's consider again Marc's case of rolling a die twice. Before you start rolling the die, the probability of getting at least one 6 is, as Marc says, 11/36. But now suppose you do your first roll of the die, and you don't get a 6. Now the only way of getting a 6 on one of the two rolls is to get a 6 on the next roll. The probability of this happening is 1/6, not 11/36. The probability of getting at least one 6 has gone down from 11/36 to 1/6, because the situation has changed as a result of the fact that you have done the first roll and you didn't get a 6. The fact that you didn't get a 6 the first time doesn't mean that the die "owes" you a 6, so the probability of getting a 6 on the second roll is no higher than if you hadn't rolled the die before.

This is related to the joke about the guy who always brings a bomb with him when he flies, because he figures that the probability of two people bringing a bomb is very low, so it's very unlikely that any of the other passengers will have a bomb. Of course, bringing a bomb with you on a plane doesn't make it any less likely that some other passenger has a bomb.

One small technical quibble with Marc's last point: When you consider an infinite sequence of coin flips, he is right that each possible outcome has probability 0. But there are uncountably many possible outcomes, so this is not a problem for countable additivity.