in a large population 65% of the people have been vaccinated, if 4 people are selected randomly what is the probability that at least 1 has been vaccinated
To find the probability that at least one person has been vaccinated out of 4 randomly selected people in a large population, we can use the concept of complementary probability.
First, let's find the probability that none of the selected individuals have been vaccinated.
Since 65% of the large population has been vaccinated, the proportion of people who have not been vaccinated is 1 - 0.65 = 0.35.
When we randomly select 4 people, the probability that each individual has not been vaccinated is 0.35. Therefore, the probability that none of the individuals have been vaccinated is:
P(No one vaccinated) = 0.35 * 0.35 * 0.35 * 0.35 = 0.35^4
Now, to find the probability that at least one person has been vaccinated, we subtract the probability that none of them have been vaccinated from 1:
P(At least one vaccinated) = 1 - P(No one vaccinated) = 1 - 0.35^4
Calculating this probability, we find:
P(At least one vaccinated) ≈ 1 - 0.35^4 = 1 - 0.0466 ≈ 0.9534
Therefore, the probability that at least one person has been vaccinated out of 4 randomly selected people is approximately 0.9534, or 95.34%.