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.

Answer 1

According to the given, 1 in every 10 women has been a victim, therefore the probability of a  victim of domestic abuse at some pint is

$p=\frac{1}{10} = 0.10$

We need to use Binomial distribution to take a randomly sampled of 25 women and asked each whether she has been a victim.

$f(x)= ^nC_x (p)^x (1-p)^{n-x}$

Where x is the event that shows a victim of domestic abuse at some point.

We need to find the probability of these victim at least 2 has been victim of abuse, so

$P(x>=2) = 1- [P(x=0)+P(x=1)]$

$P(x>=2) = 1 - [^{25}C_0(0.10)^0(1-0.10)^{25-0}+^{25}C_1(0.10)^1(1-0.10)^{25-1}]$

$P(x>=2) = 1 - [0.07198+0.199]$

$P(x>=2) = 0.72902$

So the probability is 0.729