Probability

What is defective about reasoning in the following way. Say I flip an ordinary coin and it lands heads 20 times in a row. Then I say: "Well, this combination of 20 flips is as likely as any other combination of 20 flips, so it's not so strange."

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This is an excellent question. You are right: getting heads twenty times in a row is exactly as likely as, say, HTTHTHHTTTHHTHTHHTTH.

However, 20 heads is much less likely than (say) 10 heads and 10 tails. There are many more twenty-flip combinations that yield 10 heads and 10 tails than twenty-flip combinations that yield 20 heads. There are more ways to get 10 H and 10 T than to get 20 H.

So getting 20 heads is less likely (assuming the coin is fair) than getting 10 heads and 10 tails. Yet getting 20 heads is exactly as likely as getting a particular combination of 10 heads and 10 tails, such as HTTHTHHTTTHHTHTHHTTH.

Other cases of "strangeness" are a bit more difficult to diagnose. For instance, suppose a lottery is run and ticket #1729 wins. This outcome is extremely unlikely if the lottery is fair (let's suppose there are 10,000 tickets), but extremely likely if the lottery was fixed for #1729. Does this mean that after this ticket is drawn, we should conclude that the lottery was probably fixed for #1729 (and so the outcome was not so "strange" after all)?

No! In light of the winning ticket, our confidence that the lottery is fair should reflect not only the likelihood of the outcome if the lottery is fair, but also our initial confidence (before the ticket was drawn, that is) that the lottery was fair. If there was no particular reason before the ticket was drawn for us to have been suspicious that the lottery was fixed for #1729, then the outcome will do very little to persuade us that the lottery was fixed for #1729. On the other hand, if #1729 was owned by the person running the lottery, then we might have been suspicious initially that the lottery would be fixed for #1729, in which case this ticket might count as very strong evidence that the lottery was in fact fixed for #1729.

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