Now let’s imagine you repeat that same experiment and each time you record the number of tosses required to see the desired pattern. The first time you might see HTT after 10 tosses (as in the example above), the second time you might see HTT after 7 tosses, the third time after 15 tosses, etc. After many such experiments, you calculate the average number of tosses needed to see the HTT pattern.
At the same time, imagine your friend conducts the same number of experiments but she’s looking for a different pattern: HTH (head, tail, head).
Here’s the question: on average, will it take more flips to see HTT than HTH, or vice versa, or about the same number of flips to see both patterns?
If you’re impatient, try this software simulation (which I’ve written for today’s puzzle) and the answer will reveal itself. Each test runs 1,000 trials.
Imagine you’re waiting for HTH and you see a head followed by a tail. You’re two thirds of the way there! On the next toss one of two things will happen: