Question

56% of all violent felons in the prison system are repeat offenders. If 41 violent felons are randomly selected, find the probability that a. Exactly 22 of them are repeat offenders. b. At most 23 of them are repeat offenders. c. At least 22 of them are repeat offenders. d. Between 20 and 28 (including 20 and 28) of them are repeat offenders.

Answer 1

Mean = n * P = ( 41 * 0.56 ) = 22.96
Variance = n * P * Q = ( 41 * 0.56 * 0.44 ) = 10.1024
Standard deviation = = 3.1784

part a)
P ( X = 22 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 22 - 0.5 < X < 22 + 0.5 ) = P ( 21.5 < X < 22.5 )

P ( 21.5 < X < 22.5 )
Standardizing the value

Z = ( 21.5 - 22.96 ) / 3.1784
Z = -0.46
Z = ( 22.5 - 22.96 ) / 3.1784
Z = -0.14
P ( -0.46 < Z < -0.14 )
P ( 21.5 < X < 22.5 ) = P ( Z < -0.14 ) - P ( Z < -0.46 )
P ( 21.5 < X < 22.5 ) = 0.4425 - 0.323
P ( 21.5 < X < 22.5 ) = 0.1195

Part b)

P ( X <= 23 )
Using continuity correction
P ( X < n + 0.5 ) = P ( X < 23 + 0.5 ) = P ( X < 23.5 )

P ( X < 23.5 )
Standardizing the value

Z = ( 23.5 - 22.96 ) / 3.1784
Z = 0.17

P ( X < 23.5 ) = P ( Z < 0.17 )
P ( X < 23.5 ) = 0.5675

Part c)

P ( X >= 22 )
Using continuity correction
P ( X > n - 0.5 ) = P ( X > 22 - 0.5 ) =P ( X > 21.5 )

P ( X > 21.5 ) = 1 - P ( X < 21.5 )
Standardizing the value

Z = ( 21.5 - 22.96 ) / 3.1784
Z = -0.46

P ( Z > -0.46 )
P ( X > 21.5 ) = 1 - P ( Z < -0.46 )
P ( X > 21.5 ) = 1 - 0.3228
P ( X > 21.5 ) = 0.6772

Part d)

P ( 20 <= X <= 28 )
Using continuity correction
P ( n - 0.5 < X < n + 0.5 ) = P ( 20 - 0.5 < X < 28 + 0.5 ) = P ( 19.5 < X < 28.5 )

P ( 19.5 < X < 28.5 )
Standardizing the value

Z = ( 19.5 - 22.96 ) / 3.1784
Z = -1.09
Z = ( 28.5 - 22.96 ) / 3.1784
Z = 1.74
P ( -1.09 < Z < 1.74 )
P ( 19.5 < X < 28.5 ) = P ( Z < 1.74 ) - P ( Z < -1.09 )
P ( 19.5 < X < 28.5 ) = 0.9593 - 0.1382
P ( 19.5 < X < 28.5 ) = 0.8212