Question

Suppose that about 55% of the people who are murdered actually knew the person who committed the murder. Suppose that a detective file in Boston has 51 current unsolved murders. What is the probability that fewer than 21 victims did not know their murderer?

Answer 1

24.51% probability that fewer than 21 victims did not know their murderer

Explanation:

I am going to approximate the binomial probability distribution to the normal to solve this question.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

`n = 51, p = 0.55`

So

`\mu = E(X) = np = 51*0.55 = 28.05`

`\sigma = √(V(X)) = √(np(1-p)) = √(51*0.55*0.45) = 3.55`

What is the probability that fewer than 21 victims did not know their murderer?

That is, more than 30 knew the murdered.

`P(X \geq 30 + 0.5) = P(X \geq 30.5)`, which is 1 subtracted by the pvalue of Z when X = 30.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (30.5 - 28.05)/(3.55)`

`Z = 0.69`

`Z = 0.69` has a pvalue of 0.7549

1 - 0.7549 = 0.2451

24.51% probability that fewer than 21 victims did not know their murderer