3.10: Gambler's Fallacy
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The gambler’s fallacy occurs when one thinks that independent, random events can be influenced by each other. For example, suppose I have a fair coin and I have just flipped 4 heads in a row. Erik, on the other hand, has a fair coin that he has flipped 4 times and gotten tails. We are each taking bets that the next coin flipped is heads. Who should you bet flips the head? If you are inclined to say that you should place the bet with Erik since he has been flipping all tails and since the coin is fair, the flips must even out soon, then you have committed the gambler’s fallacy. The fact is, each flip is independent of the next, so the fact that I have just flipped 4 heads in a row does not increase or decrease my chances of flipping a head. Likewise for Erik. It is true that as long as the coin is fair, then over a large number of flips we should expect that the proportion of heads to tails will be about 50/50. But there is no reason to expect that a particular flip will be more likely to be one or the other. Since the coin is fair, each flip has the same probability of being heads and the same probability of being tails—50%.