Coin flipping and the normal distribution

If the coin is flipped N times, there are 2N possible outcomes. For example, if the coin is flipped 2 times, one could find HH, HT, TH or TT, or 22=4 outcomes. If the coin if flipped 3 times, one could find HHH, HHT, HTH, THH, HTT, TTH, THT, TTT, or 23=8 outcomes.

The number of ways that we can flip the coin N times and find nH heads and nT tails is

N!/(nH! nT!)

where N! = N x (N-1) x (N-2) x ... x 2 x 1. Since the total number of possibilities is 2N, the probability of finding a particular outcome with nH heads and nT tails will be

probability(nH,nT) = (1/2N) N!/(nH! nT!)

The difference between the number of heads flipped and tails flipped is J = nH - nT. So for N flips, we can also write the probability as

probability(J) = probability(nH,nT)

The probability is least for the outcomes all heads, J=N, or all tails, J=-N. There is only one way of finding either outcome and the probability is

probability(J=N) = probability(J=-N) = 1/2N

The probability is greatest for the outcome that the number of heads is equal to the number of tails, nH = nT = N/2. There are many ways of observing such an outcome and the probability is

probability(J=0) = N!/[2N+1 (N/2)! ]

As the number of flips N becomes very large, the number of heads flipped will be nearly equal to the number of tails flipped. The probability distribution will be well approximated by a normal distribution

probability(J) = (2 π N)-1/2 exp[-J2/2N]

which is also known as a "bell curve" or a "gaussian distribution." This distribution can be used to describe many interesting situations, from the probability of observing a particular sequence of coin flips, to the distribution of darts hitting a dart board, to the probability of observing particular speeds of atoms and molecules.

A very important property of the distribution is that it has a width that is proportional to the square root of N. While the number of flips increases as N, the width of the distribution increases only as N1/2.