Question

According to a recent study, 1 in every 10 women has been a victim of domestic abuse at some point in her life. Suppose we have randomly and independently sampled twenty-five women and asked each whether she has been a victim of domestic abuse at some point in her life. Find the probability that at least 2 of the women sampled have been the victim of domestic abuse. Round to six decimal places.

Answer 1

The probability that at least 2 out of 25 women sampled have been victims of domestic abuse is approximately 0.9296, or 92.96%.

To find the probability that at least 2 out of 25 women sampled have been victims of domestic abuse, we can use the complement probability. The complement probability is the probability of the event not happening subtracted from 1.

Let's denote the probability that a randomly selected woman has been a victim of domestic abuse as p. According to the study,

p=
`(1)/(20)=0.1`

The probability that none of the women have been victims of domestic abuse is
`(1-p)^{25`

So, the probability that at least 2 women have been victims is
`1-(1-p)^{25`

Let's calculate it:

P(at least 2 women have been victims) = 1
`(10.1)^{25`

P(at least 2 women have been victims) = 1
`(0.9)^{25`

P(at least 2 women have been victims) ≈ 1-0.07039

P(at least 2 women have been victims) ≈ 0.9296

So, the probability that at least 2 out of 25 women sampled have been victims of domestic abuse is approximately 0.9296, or 92.96%.