Question

Statistics is proving to be my nemesis!

According to Harper's Index, 55% of all federal inmates are serving time for drug dealing. A random sample of 15federal inmates is selected.

(a) What is the probability that 8or more are serving time for drug dealing? (Round your answer to three decimal places.)

(b) What is the probability that 2or fewer are serving time for drug dealing? (Round your answer to three decimal places.)

(c) What is the expected number of inmates serving time for drug dealing? (Round your answer to one decimal place.)

Answer 1

X : Number of federal inmates serving time for drug dealing.

Probability  that an inmate serving time for drug dealing : p = 55/100 = 0.55

q = 1-p = 1-0.55 = 0.45

n : number of federal inmates selected randomly = 15

X follows Binomial distribution with n= 15 and p = 0.55. And the probability mass function.

Probability that 'r' of the 15 federal inmates serving time for drug dealing P(X=r) is given by

(a) Probability that 8 or more are serving time for drug dealing = P(X8)

x	P(x)	P(x)
8		0.201344
9		0.191401
10		0.14036
11		0.077978
12		0.031769
13		0.00896
14		0.001565
15		0.000127
	Total	0.653504

Probability that 8 or more are serving time for drug dealing = P(X8) = 0.654

(b) Probability that 2 or fewer are serving time for drug dealing = P(X 2) = P(X=0)+P(X=1)+P(X=2)

x	P(x)	P(x)
0		0.000006
1		0.000115
2		0.000986
	Total	0.001107

Probability that 2 or fewer are serving time for drug dealing = 0.001

(c) Expected number of inmates serving time for drug dealing

E(X) of Binomial distribution = np

Expected number of inmates serving time for drug dealing = E(X) = np = 15 x 0.55 = 8.25

Expected number of inmates serving time for drug dealing = 8.3

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Binomial distribution:

Binomial Distribution

X : Follows binomial distribution

If 'X' is the random variable representing the number of successes, the probability of getting ‘r’ successes and ‘n-r’ failures, in 'n' trails, ‘p’ probability of success ‘q’=(1-p) is given by the probability function