Suppose you’ve seen a coin come up heads 10 times in a row. What do you believe is likely to happen next? Three common responses:
- Equal probability of heads or tails.
Each is reasonable in its own context. The last answer is correct assuming the flips are independent and heads and tails are equally likely.
But as I argued here, if you see nothing but heads, you have reason to question the assumption that the coin is fair. So there’s some justification for the first answer.
The reasoning behind the second answer is that tails are “due.” This isn’t true if you’re looking at independent flips of a fair coin, but it could reasonable in other settings, such as sampling without replacement.
Say there are a number of coins on a table, covered by a cloth. A fixed number are on the table heads up, and a fixed number tails up. You reach under the cloth and slide a coin out. Every head you pull out increases the chances that the next coin will be tails. If there were an equal number of heads and tails under the cloth to being with, then after pulling out 10 heads tails are indeed more likely next time.
Related post: Long runs