Question

56% 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 21 of them are repeat offenders.
c. At least 18 of them are repeat offenders.  
d. Between 15 and 20 (including 15 and 20) of them are repeat offenders.

Answer 1

solution:-

given that p = 0.56 , n = 34

binomial formula

=> P(x = X) = ncr * p^r * (1-p)^(n-r)

a. Exactly 20 of them are repeat offenders.

P(x = 20) = 34c20 * 0.56^20 * (1-0.56)^(34-20)

= 0.1305

b. At most 21 of them are repeat offenders

P(x ≤ 21) = 0.8016

c. At least 18 of them are repeat offenders

P(x ≥ 18) = 0.7041

d. Between 15 and 20 (including 15 and 20) of them are repeat offenders.

P(15 < x < 20) = P(x = 15) + P(x = 16) + P(x = 17) + P(x = 18) + P(x = 19) + P(x = 20)

= 34c10 * 0.56^15 * (1-0.56)^(34-15) + 34c16 * 0.56^16 * (1-0.56)^(34-16) + 34c17 * 0.56^17 * (1-0.56)^(34-17) + 34c18 * 0.56^18 * (1-0.56)^(34-18) + 34c19 * 0.56^19 * (1-0.56)^(34-19) + 34c20 * 0.56^20 * (1-0.56)^(34-20)

= 0.0521 + 0.0788 + 0.1061 + 0.1276 + 0.1368 + 0.1305

= 0.6319