Question

In a​ study, 4040​% of adults questioned reported that their health was excellent. A researcher wishes to study the health of people living close to a nuclear power plant. Among 1212 adults randomly selected from this​ area, only 3 reported that their health was excellent. Find the probability that when 1212 adults are randomly​ selected, 3 or fewer are in excellent health. Round to three decimal places.

Answer 1

0.225 is the probability that 3 or fewer than 3 adults are in excellent health.

Explanation:

We are given the following information:

We treat adult with excellent health as a success.

P(Adult with excellent health) = 40% = 0.4

Then the number of adults follows a binomial distribution, where

`P(X=x) = \binom{n}{x}.p^x.(1-p)^(n-x)`

where n is the total number of observations, x is the number of success, p is the probability of success.

Now, we are given n = 12

We have to evaluate:

`P(x \leq 3)\\ = P(x = 0) + P(x = 1) + P(x=2) + P(x=3) \\= \binom{12}{0}(0.4)^0(1-0.4)^(12) +\binom{12}{1}(0.4)^1(1-0.4)^(11) +\binom{12}{0}(0.4)^2(1-0.4)^(10)\\+\binom{12}{0}(0.4)^3(1-0.4)^(9)\\= 0.0021 + 0.0174 + 0.0638 + 0.1418\\= 0.225`

0.225 is the probability that 3 or fewer than 3 adults are in excellent health.