a)0.783
b)0.409
c) 0.024
d)1.422
e) σ = 1.09
Step - by - Step Explanation
What to find?
• Find the probability that one or more of the five parolees will be repeat offenders.
• Find the probability that two or more of the five parolees will be repeat offenders.
• Find the probability that four or more of the five parolees will be repeat offenders.
• Compute μ, the expected number of repeat offenders out of five
,• Compute σ, the standard deviation of the number of repeat offenders out of five
Given Parameters:
x 0 1 2 3 4 5
P(x) 0.217 0.374 0.204 0.181 0.023 0.001
a) Probability that one or more of the five parolees will be repeat offenders.
P(x≥ 1) = 1 - p(x < 1)
= 1 - P(x=0)
= 1 - 0.217
= 0.783
OR
P(x≥1) = p(x=1) + p(x=2) + p(x=3) + p(x=4) + p(x=5)
= 0.374 + 0.204 + 0.181 + 0.023 + 0.001
=0.783
Hence, Probability that one or more of the five parolees will be repeat offenders is 0.783
This is the complement of the probability of no repeat offenders as shown above.
Hence, the first option is the correct answer.
b) The probability that two or more of the five parolees will be repeat offenders.
P(x≥ 2) = p(x=2) + p(x=3) + p(x=4) + p(x=5)
= 0.204 + 0.181 + 0.023 + 0.001
=0.409
Hence, probability that two or more of the five parolees will be repeat offenders is 0.409
c)The probability that four or more of the five parolees will be repeat offenders.
P(x≥4) = p(x=4) + p(x=5)
= 0.023 + 0.001
= 0.024
Hence, the probability that four or more of the five parolees will be repeat offenders is 0.024
d)Compute μ, the expected number of repeat offenders out of five
To compute μ, multiply each x by its corresponding P(x) and sum it up.
That is;
Mean = E(X) = Σ x * P(x)
=0(0.217) + 1(0.374) + 2(0.204) + 3(0.181) + 4(0.023) + 5(0.001)
= 0 + 0.374 + 0.408 + 0.543 + 0.092 + 0.005
= 1.422
Hence, the expected number of repeat offenders out of five is 1.422
e)Compute σ, the standard deviation of the number of repeat offenders out of five
To compute σ,
compute E(x²) = Σ x² * p(x)
= 0²(0.217) + 1²(0.374) + 2²(0.204) + 3²(0.181) + 4²(0.023) + 5²(0.001)
= 0(0.217) + 1(0.374) + 4(0.204) + 9(0.181) + 16(0.023) + 25(0.001)
= 0 + 0.374 + 0.816 + 1.629 + 0.368 + 0.025
= 3.212
Variance = Σ x² *p(x) - [Σ x *p(x) ] ²
= 3.212 - 1.422²
= 3.212 - 2.022084
=1.189916
Standard deviation (σ) = √ 1.189916 = 1.09
Hence, σ = 1.09
