Chapter 15 The Binomial Formula

15.1 Chapter Notes

The chapter introduces the binomial formula:

The chance that an event will occur exactly k times out of n is given by the binomial formula

\[ \frac{n!}{k!(n-k)!} p^k (1-p)^{n-k} \]

In this formula, n is the number of trials, k is the number of times the event is to occur, and p is the probability that the event will occur on any particular trial. The assumptions:

• The value of n must be fixed in advance.
• p must be the same from trial to trial.
• The trials must be independent.

The binomial coefficient \(\frac{n!}{k!(n-k)!}\) is the number of ways to choose \(k\) objects from a set of \(n\) if you don’t care about order.

The formula \(p^k(1-p)^{1-k}\) is the probability of any particular choice of \(k\) objects.

To get the total probability that \(k\) objects are chosen, you use the addition rule to add up the probabilities of all of the possible choices. I.e. you add \(p^k(1-p)^{1-k}\) to itself \(\frac{n!}{k!(n-k)!}\) times, which is where the formula comes from.

Alternative notation for the binomial coefficient includes \(n \choose k\) and \(^nC_k\). Read “n choose k.”