need to know the formula ?

Question

In a large population, 60% of the people have been vaccinated. If 3 people are randomly selected, what is the probability that AT LEAST ONE of them has been vaccinated?

Give your answer as a decimal to 4 places.

Answer 1

This is a Binomial experiment with a probability of success p=0.6. n=3

PMF for Binomial distribution is P(X=x)=n!/[x!(n-x)!]*p^x(1-p)^n-x

Probability of at least one success in three trials is P(at least one)=1-P(No vaccinated) or P(at least one)=P(1 vaccinated)+P(2 vaccinated)+P(3 vaccinated).

P(X≥1)=1-P(X<1)=1-P(X=0)

P(X=0)=3!/[0!(3-0)!]*0.6⁰(1-0.6)^3-0=0.4³=0.064 → P(X≥1)=1-P(X=0)=1-0.0640=0.9360

or

P(X=1)= 3!/[1!(3-1)!]*0.6¹(1-0.6)^3-1=3*0.6*0.4²=0.2880

P(X=2)=3!/[2!(3-2)!]*0.6²(1-0.6)^3-2=3*0.6²*0.4=0.4320

P(X=3)=3!/[3!(3-3)!]*0.6³(1-0.6)^3-3=0.6³=0.2160

P(X≥1)=P(X=1)+P(X=2)+P(X=3)=0.2880+0.4320+0.2160=0.9360