Question

59% of all violent felons in the prison system are repeat offenders. If 37 violent felons are randomly selected, find the probability that  

a. Exactly 22 of them are repeat offenders. ____
b. At most 22 of them are repeat offenders. ____
c. At least 20 of them are repeat offenders. ____  
d. Between 20 and 24 (including 20 and 24) of them are repeat offenders. ____

* Enter an integral or decimal number.

Answer 1

it is a binomial probability distribution,      
      
      
and probability is given by      

P(X=x) = C(n,x)*px*(1-p)(n-x)

where  
Sample size , n =    37
Probability of an event of interest, p =   0.59

a)

P ( X =    22   ) = C(   37   ,   22   )*   0.59   ^   22   *   0.41   ^   15   =   0.1324(answer)

b)

P(X≤22) = Σ C(n,x)*p^x*(1-p)^(n-x) where x goes from 0 to 22

P(X≤22) = 0.5846

c)

P(X≥20) = 1 - P(X<19) = 0.2171(answer)

d)

P(20≤X≤24) = P(X≤24) - P(X<20) = 0.5962(answer)