Coin Tosses

The behavior of many systems and materials can be better understood by considering the random walk of particles in the system. To get some “feeling” for a random walk, consider the following exercise. Divide a group of people into two groups. Have each individual in one group toss a fair coin 256 times and write down in sequence the outcome of each toss. Have each individual in the other group write down what they would imagine a typical sequence of 256 random tosses to be but not actually do the tossing. Collect all the papers and mix them up thoroughly. Can you determine with reasonable accuracy which sets of data were obtained experimentally? How accurate should your selection be?

Answer

You should be able to pick out the experimentally obtained sequences with about 98 percent accuracy! In a random sequence of 256 fair coin tosses, you would expect to find at least 1 run of 6 heads or 6 tails with a probability of 98.2 percent. If the sequences imagined by students unfamiliar with the characteristics of randomness do not contain long runs, you should be able to distinguish them reliably.

The actual estimate of the number of runs with 6 or more heads or tails is 4, meaning that you should be able to find about 4 of these long runs. For a run of at least k heads in n tosses, where k ≥ 1, the mean number of runs is ~ n/2^(k+1); thus 2 (256/2⁷) = 4. The following table contains actual data for 256 coin tosses, with a 1 representing heads. You can count the numbers of the different run lengths.

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