Question

20% of the people in a community use the emergency room at in one year,find the probability that at most three used the emergency room for a sample of 10 people

Answer 1

The probability that at most three used the emergency room for a sample of 10 people is 0.8791.

Explanation:

The random variable X can be defined as the number of people in a community using the emergency room.

The probability that a person uses the emergency room is, p = 0.20.

A sample of n = 10 people are selected.

A person using the emergency room is independent of others.

The random variable X thus follows a Binomial distribution with parameters n = 10 and p = 0.20.

The probability mass function of X is provided as follows:

`P(X=x)={10\choose x}\ (0.20)^(x)\ (1-0.20)^(10-x);\ x=0,1,2,3...`

Compute the probability that at most three used the emergency room as follows:

P (X ≤ 3) = P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3)

`=\sum\limits^(3)_(x=0){{10\choose x}\ (0.20)^(x)\ (1-0.20)^(10-x)}\\\\=0.10737+0.26844+0.30199+0.20133\\\\=0.87913\\\\\approx 0.8791`

Thus, the probability that at most three used the emergency room for a sample of 10 people is 0.8791.