From recent polls about customer satisfaction, you know that 85% of the clients of your company are highly satisfied and want to renew their contracts. If you select a random sample of 30 clients, what is the probability that at least one client is dissatisfied?

99.24% probability that at least one client is dissatisfied

Step-by-step explanation:

For each client, there are only two possible outcomes. Either they are dissatisfied, or they are not. The probability of a client being dissatisfied is independent from other clients. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

In which is the number of different combinations of x objects from a set of n elements, given by the following formula.

And p is the probability of X happening.

85% of the clients of your company are highly satisfied

So 15% are dissatisfied, so

If you select a random sample of 30 clients, what is the probability that at least one client is dissatisfied?

This is when

We know that either no clients are dissatisfied, or at least one is. The sum of the probabilities of these events is decimal 1. So

In which

99.24% probability that at least one client is dissatisfied

x = 12

y = 1

for the points!