Question

13. According to Harper’s Index, 50% of all federal inmates are serving time for drug dealing. A random sample of 16 federal inmates is selected. (a) What is the probability that 12 or more are serving time for drug dealing? (b) What is the probability that 7 or fewer are serving time for drug dealing? (c) What is the expected number of inmates serving time for drug dealing?

Answer 1

Answer:

Given,

n = 16

p = 50% = 0.5

q = 1 - 0.5

q = 0.5

a)

To determine the probability that 12 or more are serving time for drug dealing

P(x >= 12) = P(x = 12) + P(x = 13) + P(x = 14) + P(x = 15) + P(x = 16)

= 16C12*0.5^12*0.5^4 + 16C13*0.5^13*0.5^3 + 16C14*0.5^14*0.5^2 + 16C15*0.5^15*0.5 + 16C16*0.5^16*0.5^0

= 0.0278 + 0.0085 + 0.0018 + 0.0002 + 0

= 0.0383

b)

To determine the probability that 7 or fewer are serving time for drug dealing

P(x <= 7) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6) + P(7)

= 16C0*0.5^0*0.5^16 + 16C1*0.5^1*0.5^15 + 16C2*0.5^2*0.5^14 + 16C3*0.5^3*0.5^13 + 16C4*0.5^4*0.5^12 + 16C5*0.5^5*0.5^11 + 16C6*0.5^6*0.5^10 + 16C7*0.5^7*0.5^9

= 0 + 0.0002 + 0.0018 + 0.0085 + 0.0278 + 0.0667 + 0.1222 + 0.1746

= 0.4018

c)

To determine the expected number of inmates serving time for drug dealing

Mean = E(X) = np

= 16*0.5

= 8

Standard deviation = sqrt(npq)

= sqrt(16*0.5*0.5)

= 2