Strong Theorems on Coin Tossing

“The generation of random numbers is too important to be left to chance.”
– Robert R. Coveyou

In the paper “Strong theorems on coin tossing,” mathematician Pál Révész tells a story about a high-school class. The teacher leaves the room, and the children are told to divide into two groups.

In the first group, students have to flip a coin two hundred times and record whether a flip lands heads or tails. In the other group, students have to come up with random heads-or-tails sequences, but without using an actual coin; instead, they have to “generate” random numbers in their head.

Once all the students each have 200 lines recorded, the teacher returns and tries to guess which student belongs to which group. Most of the time, the teacher guesses quite well.

Their secret is this – for the average person, it doesn’t feel right to put down five, or even four, consecutive heads or tails. That many runs just doesn’t sound plausible. However, a statistician knows that when a uniformly random coin generates a sequence of 200, it’s very likely to have runs of six or more. Its probability is close to 97 percent.