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What if we flip a biased coin, with the probability of a head p and the probability of a tail q = 1 - p? The probability of a given sequence, e.g., 100010 ..., in which k heads appear in n flips is, by Eq. A.4, pqqqpq ..., or

Equation A.16 (A.16)

There are a total of 2n possible sequences. Only some of these give k heads and n - k tails. Their number is

Equation A.17 (A.17)

where 0!, whenever it appears in the denominator, is understood to be 1.

Since any one or another of these sequences will do, the probability that exactly k heads occur in n flips is, by Eq. A.2.

Equation A.18 (A.18)

This is the binomial distribution. The coefficient n choose k is the binomial coefficient, the number of combinations of n things taken k and n - k at a time. You have seen it before in algebra in the binomial theorem:

Equation A.19 (A.19)

We can use the binomial theorem to show that the binomial distribution is normalized:

Equation A.20 (A.20)

As an example, let's work out the case of 4 flips of an unbiased coin. If p = q = 1/2, then pkqn-k = (1/2)n = (1/2)4 = 1/16 for all values of k, and the probabilities P(0;4,1/2), ..., P(4;4,1/2) are equal to the binomial coefficients 4 choose 0, ... , 4 choose 4 times this factor. Since




we obtain the probabilities 1/16, 1/4, 3/8, 1/4, and 1/16, as before.

The expectation value of k is

Equation A.21 (A.21)

To evaluate this, note that the k = 0 term is 0 and that k/k! = 1/(k - 1)!, so that


Next, factor out np:


Finally, change variables by substituting m = k - 1 and s = n - 1:


The sum in this expression is the same as the one in Eq. A.20; only the labels have been changed. Thus,

Equation A.22 (A.22)

One can evaluate the expectation value of k2 in a similar fashion by two successive changes in variables and show that

Equation A.23 (A.23)

The variance of k, Eq. A.14, is

Equation A.24 (A.24)

and its standard deviation is

Equation A.25 (A.25)

An example of the binomial distribution is given in Fig. A.4, which shows the theoretical distribution P(k;10,1/6). This is the probability of obtaining a given side k times in 10 throws of a die.

Figure 4

Figure A.4. The binomial distribution for n = 10, p = 1/6. The mean value is 1.67, the standard deviation 1.18.

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