64% of all violent felons in the prison system are repeat offenders. If 34 violent felons are randomly selected, find the probability that

Question

64% of all violent felons in the prison system are repeat offenders. If 34 violent felons are randomly selected, find the probability that  

a. Exactly 20 of them are repeat offenders. ?

b. At most 24 of them are repeat offenders. ?

c. At least 20 of them are repeat offenders. ?

d. Between 17 and 21 (including 17 and 21) of them are repeat offenders. ?

Answer 1

Hi Jenessa,

This is the same probability type, formula, and procedure as the other problem you posted about bird survival.

Binomial Probability Formula:

P(X=x) = C(n,x)*p^x*q^n-x

x= Desired number of repeat offenders = 20 for part a

n= Total number of offenders = 34, all parts

p = Probability of repeat offender = 0.64 for all parts

q= Probability of non-repeat offender = 0.36 for all parts

C(n,x) = n!/x!(n-x)!; !="Factorial" i.e 4*3*2*1--you will need a calculator or software for this

a. P(X=20) = C(34, 20)*0.64²⁰*0.36¹⁴ = 0.114

b. You want at most 24 repeat offenders, so you need probability for everything 24 and below. This is a lot of work. Instead, take probabilities for all x-values above 24 and apply the Complement Rule aka the "One Minus Trick."

P(X=25) = C(34, 25)*0.64²⁵*0.36⁹

P(X=26) = C(34, 26)*0.64²⁶*0.36⁸

P(X=27) = C(34, 27)*0.64²⁷*0.36⁷

^....

Continue this until you reach 34, then add all probabilities together. This gives the probability of over 24 repeat offenders P(X>24).

Finally, apply the Complement Rule: P(X<=24) = 1 - P(X>24)

c. Same procedure without Complement Rule since you are interested in at least 20:

P(X=20) = C(34, 20)*0.64²⁰*0.36¹⁴

P(X=21) = C(34, 21)*0.64²¹*0.36¹³

P(X=22) = C(34, 22)*0.64²²*0.36¹²

....

Repeat until you reach X=34; note that you computed some of these in previous parts. Add all probabilities together to get P(X>=20).

d. Same procedure.

P(X=17) = C(34,17)*0.64¹⁷*0.36¹⁷

P(X=18) = C(34,18)*0.64¹⁸*0.36¹⁶

P(X=19) = C(34,19)*0.64¹⁹*0.36¹⁵

^....

Continue until you reach X=21; add the probabilities together, and you will have the probability of between 17 and 21 repeat offenders. I hope this helps.