In a large population, 56 % of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Question

In a large population, 56 % of the people have been vaccinated. If 5 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Answer 1

Use the binomial probability distribution since the question of vaccinated or unvaccinated is a yes/no or dichotomous outcome. Then, instead of calculating the probabilities of exactly 1, 2, 3, 4, and all 5 selected people are vaccinated, calculate the probability that 0 are vaccinated and subtract this probability from 1.

P₀ = n!/[(n - k)!k!]p^kq^n-k



5!/[(5 - 0)!0!](0.56)⁰(0.44)^5-0 = (1)(1)(0.0165) = 0.0165

P_~0 = 1 - P₀ = 1 - 0.0165 = 0.9835