Question

According to a recent study, 1 in every 9 women has been a victim of domestic abuse at some point 19) 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.
1. 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

Answer:

$P(X\geq 2)=1-P(X\leq 1)=1-[0.054294+0.167762]=0.777944$

Step-by-step explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:  

$P(X)=(nCx)(p)^x (1-p)^{n-x}$  

Where (nCx) means combinatory and it's given by this formula:  

$nCx=\frac{n!}{(n-x)! x!}$  

The complement rule is a theorem that provides a connection between the probability of an event and the probability of the complement of the event. Lat A the event of interest and A' the complement. The rule is defined by: $P(A)+P(A') =1$

Find the probability that at least 2 of the women sampled have been the victim of domestic abuse.

On this case we want to find this probability

$P(X\geq 2) =1-P(X$

And we can find the individual probabilities like this:

$P(X=0)=(25C0)(0.11)^0 (1-0.11)^{25-0}=0.054294$  

$P(X=1)=(25C1)(0.11)^1 (1-0.11)^{25-1}=0.167762$  

$P(X\geq 2)=1-P(X\leq 1)=1-[0.054294+0.167762]=0.777944$