Question

Twenty percent of the clients of a large hair salon are female. In a random sample of 4 clients, what is the probability that exactly 3 clients are female?

Answer 1

`4\cdot (0.2)^3\cdot 0.8`

Explanation:

We may be tempted to list all possible outcomes, and compute their probabilities: after all, if we must sample `3` females `F` out of four clients, the possibilities are

`(F,F,F,M), (F,F,M,F), (F,M,F,F), (M,F,F,F)`

Each client is female with probability `0.2`, and thus male with probability `0.8`. So, each quadruplet we just wrote has probability

`0.2\cdot0.2\cdot0.2\cdot0.8 = (0.2)^3\cdot 0.8`

Since we have four events with such probability, the answer will be

`4\cdot (0.2)^3\cdot 0.8`

But what if the numbers were much greater? Listing all possible events would quickly become cumberstone. That's why we have models: this situation is described by a bernoullian model, which means that if we want to achieve `k` successes in `n` experiments with probability of success `p`, then our probability is

`P=((n),(k))p^k(1-p)^{n-k}`

where

`((n),(k)) = \frac{n!}{k!(n-k)!}` and `n! = n(n-1)(n-2)...3\cdot2`

In this case, `n=4`, `k=3` and `p=0.2`, so

`P=((4),(3))0.2^3(0.8) =4\cdot0.2^3(0.8)`