Binomial Distribution – Equation

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Last time we looked at the binomial distribution it was purely to understand the behavior of a random variable. This time we’re going to look at the equation that defines it.

Here it is:

\displaystyle \binom{n}{k}p^k(1-p)^{n-k}

All of this notation should by now be familiar. The first part is a binomial coefficient and the rest is basic math. The variable n is the sample size and the variable p is the probability of success. Meanwhile k is the set of natural numbers N0 (whole numbers from 0 to infinity). We test the probability at each value of k in order to create the distribution.

So why is this the equation?

I am not in the business of writing proofs, many good ones exist, but rather of producing intuition. The binomial coefficient is there because we are choosing from a set. It is there as a normalizing value to convert the rest of the equation into a meaningful probability.

Let us flips one coin three times. The probability that the coin will come up heads three times is p^3 since it has probability p each time. Alternatively the exponential term could be expanded to p*p*p. When we plug this into the equation:

\displaystyle \binom{3}{3}p^3(1-p)^{3-3}

Which can then be solved to become:

1\times p^3(1-p)^0
. . . which is . . .
1\times p^3\times1
. . . which is . . .
p^3

So it gives us the right answer.

How about the probability of getting two heads? The probability of getting heads twice is p^2 and the probability of getting one tails is (1-p)^1 so we multiply them together to find the probability. The result can be expanded to p*p*(1-p). This is not the probability of getting two heads and a tails, however, it is the total probability of getting each result.

What is the difference?

There are three equally likely ways to get this result so we have a probability of (p^2)*(1-p) three times! The equation is:

\displaystyle \binom{3}{2}p^2(1-p)^{3-2}

Which can then be solved to become:

\displaystyle 3\times p^2(1-p)^{1}

Exactly what we predicted.

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