Question

The three-­year recidivism rate of parolees in Texas is 30% (www.lbb.state.tx.us). In other words, 30% of released prisoners return to prison within three years of their release. Suppose a prison in Texas released 15 prisoners. (Problem taken from “Introductory Statistics” by robert Gould). Assuming that whether one prisoner returns to prison is independent of whether any others return.

1. What is the probability that exactly 5 will return to prison?

2. What is the probability that more than 5 will return to prison?

3. What is the probability that less than 5 will return to prison?

Do you notice anything about the answers to these three parts? (Hint: consider their sum)

4. What is the probability that no one goes back to prison?

5. What is the probability that at least one person goes back to prison?

Do you notice anything about the relationship between these two answers?

Do you notice anything about the answers to these three parts? (Hint: consider their sum)

4. What is the probability that no one goes back to prison?

5. What is the probability that at least one person goes back to prison?

Do you notice anything about the relationship between these two answers?

Answer 1

1. Let X be the number of person return to prison

X follow Binomial with probability p= 0.30 , n=15

The probability mass function of X is

To find P(X=5)

Probability that exactly 5 will return to prison = 0.2061

2. To find P(X>5)

Probability that more than 5 will return to prison = 0.2784

3. To find P(X<5)

Probability that less than 5 will return to prison = 0.5155

We have found

P(X=5) +P(X>5) +P(X<5) = 1 (total probability )

Thus , we can say that the three cases , X=5 , X> 5 and X<5 are exhaustive number of cases.

4. To find P(X=0)

Probability that no one will return to prison = 0.0047

5.  To find P(X1)

Probability that at least one will return to prison = 0.9953

We have found that

P(X=0) + P(X 1) = 0.9953 (total probability)

Thus , we can say that the two cases , X=0 , X1 exhaustive number of cases.

Note :We can find the probabilities using excel function as follows

P(X5 ) = "=BINOM.DIST(5,15,0.3,cumulative )" = 0.7216

P(X4 ) = "=BINOM.DIST(4,15,0.3,cumulative )" = 0.5155