Leslie's Probability

Leslie’s intuitions about probability seem to be of a kind with the case of the double lottery winner. Suppose that you buy lottery tickets for Tuesday and Friday. There are on both occasions ten million possible numbers, so your chances of winning on Tuesday or Friday are each one in ten million. And now suppose that you win on Tuesday. What are your chances that you will win on Friday also? Most people will feel that the chances of winning on Friday after having won on Tuesday must be far less than your chances when you won on Tuesday: one in ten million times ten million (1/10¹⁴). But this is a mistake. It is one thing to ask, what are your chances that you will win in both Tuesday and Friday? The answer is: 1/10¹⁴. It is another thing to ask, what are the chances that you will win on Friday after winning on Tuesday? Well, you have one ticket and there are 10 million possible numbers. So your chances should be one in ten million (10⁷). But people like Leslie find this hard to accept. They think that there is something valid about the intuition that tells us that after winning the lottery once, the chances of winning it again have to be far smaller than they were the first time. The difficulty comes from thinking, in this case, that if you won on Tuesday, to win again on Friday really means that you will have won on both Tuesday and Friday and that should have an extremely small probability (even compared to the small probability of winning one lottery to begin with).

To dispel this intuition, let us look at one example in which the closer we get to an unlikely combined result, the greater its probability. Suppose you flip a fair coin a hundred times. What are the chances that you will get a hundred heads? 1/2¹⁰⁰. That is an extremely small number. Suppose now that on the first flip of the coin you get heads. What are the chances now that you will get 100 heads? There are 99 flips left. That means that you will have to get heads on all 99. Your chances then will be 1/2⁹⁹. That is a very small number, too, but it is twice as large as the number you had before (because the denominator, 2^99, is half of 2¹⁰⁰: 2x2⁹⁹ is 2¹⁰⁰). After getting 2 heads, your chances of getting 100 heads will be 1/2⁹⁸, which is a very small number, but twice as large as that of your second flip and 4 times as large as your first’s. The more heads you get, the closer you get to getting 100 heads, the more your chances improve of getting 100 heads. Suppose you have flipped 98 heads, so you have only two flips left. Your chances will be 1/2²=1/4. You have gotten 99 heads? Your chances of getting 100 heads depend on your getting heads on your last try. That is 1/2.

Of course, the chances of getting 99 heads and one tail will at that point also be 1/2. At the start, there were 2¹⁰⁰ possible combinations; getting 100 heads was just one of them (1/2¹⁰⁰). But once you got heads on your first try, you eliminated half the possible combinations, that is 2⁹⁹ hitherto possible combinations. Check it: 2⁹⁹ + 2⁹⁹ = 2¹⁰⁰. For an easier example: 2² + 2² = 2³ (4 + 4 = 8) and 2³ + 2³ = 2⁴ (8 + 8 = 16). If your base were 3, then you would have to add three numbers: 3^{2 +} 3² + 3² = 3³ (9 + 9 + 9 = 27). If your base were 4 you would have to add four numbers, and so on. Now, when you get your second head in a row, you eliminate half of the existing possible combinations left up to that point, that is, you eliminate 2⁹⁸ additional possible combinations. And so on: every time you get heads, you eliminate half of the possible combinations still in existence up to that flipping of the coin. Until, finally, when you only have one flip to go, there are only two possible combinations left. The first will give you 99 heads and then tails. The second will give you100 heads as the result of getting heads on your last flip. Both combinations are equally probable. Therefore, your chances of getting 100 heads when you have already gotten 99 are far greater than when you first got a head, while the chances of getting a head on your last flip are exactly the same as the chances of getting a head on your first flip: 1/2.

Let us remember this reasoning when we begin to think about the probability that life could evolve from inorganic materials.