Imagine you’re bored one day and decide to start flipping a coin just to pass the time. The first flip lands on heads. The second flip also lands on heads. Then the third. That’s a little unlikely. To your surprise, the fourth flip also lands on heads.
At this point, you pause to consider the probabilities of your next flip. On one hand, you know that each coin flip is an independent even with a probability of landing on heads of 1/2. The odds of flipping a fifth heads in a row should be 1 in 2, then. On the other hand, you also know that a streak of five heads in a row has a probability of 1/32. That’s a lot less than 1 in 2, and only half as likely to occur as a streak of four heads in a row. So it would also seem that the odds of landing a fifth heads is 1 in 32.
What’s going on here?
The mistake: ignoring the probability of tails
Counterintuitively, the probability is in fact 1/2. More specifically, the sequence HHHHH has the same probability as the sequence HHHHT. Every particular sequence of five coin flip outcomes has the same probability (all 32 of them).
There are a couple of factors behind this error. A sequence of all heads is more significant to the human mind than a(n equally likely) mixed sequence of heads and tails, but that’s not all that’s going on. Note that HHHHH is genuinely less likely in the sense that, out of 32 possible outcomes, only two of them are the same result repeated five times (HHHHH and TTTTT). This means that the probability of five of the same outcome is 2/32 or 1/16. Once you flip the first heads, the possibility of five tails in a row is precluded, so you might expect the probability to become 1/32.
The problem with this perspective is that it mixes up prospective and retrospective probabilities. For example, after you’ve flipped the first heads, the probability of five heads in a row actually becomes 1/16, because from that point on you only need to flip four heads in a row. And, more to the point, by the time you’ve already flipped four heads, the probability of completing the sequence is 1/2. As you flip each coin, you eliminate half the remaining possible outcomes for the full five-flip sequence.
So while flipping five heads in a row is half as likely as flipping four heads in a row, the fifth heads is exactly as likely as the fourth. The “half as likely” is exactly accounted for by the fifth heads having a probability of 1/2: in half of the cases we flip four heads in a row, our fifth flip will be heads.
Another way of looking at this is that all past outcomes have a probability of 1, so once HHHH has been flipped the probability of HHHHH is 1*1*1*1*1/2 = 1/2.
See also: Why do unlikely things happen?
Photo by Pixabay, edited by me (CC0 Public Domain)
thoughts on mind 