Answer:
a) P(X=5) = 0.065
b) P(X≥1) = 0.928
c) The mean number of repeat offenders is 3.
d) The standard deviation of the number of repeat offenders is 2.
Explanation:
n = 25
p = 10% = 0.10
q = 1-p
= 1-0.1
q=0.9
a) Find P(X =5)
Using the binomial distribution formula:
P(X = x) = ⁿCˣ pˣ qⁿ⁻ˣ
where p = probability of success
q = probability of failure
n = total number of trials
x = no. of successful trials
P(X=5) = ²⁵C₅ (0.1)⁵ (0.9)²⁵⁻⁵
= 0.0645 ≅ 0.065
P(X=5) = 0.065
b) We need to find the probability that at least one person is a repeat offender which means at most there can be 25 repeat offenders hence, we need to compute the probability for all values ranging from 1 to 25. The easy way to solve this problem is to find out the probability of less than one repeat offender and then subtract the value from the total probability (i.e. 1).
P(X≥1) = 1 - P(X<1)
= 1 - P(X =0)
= 1 - ²⁵C₀ (0.1)⁰ (0.9)²⁵⁻⁰
= 1 - 0.072
P(X≥1) = 0.928
c) The mean of the binomial distribution is:
μ = np
So, the mean number of repeat offenders can be computed as:
μ = (25)(0.10)
μ = 2.5 ≅ 3
The mean number of repeat offenders is 3.
d) The standard deviation of the binomial distribution is:
σ = √npq
So, the standard deviation of the number of repeat offenders can be computed as:
σ = √(25)(0.1)(0.9)
= √2.25
σ = 1.5 ≅ 2
The standard deviation of the number of repeat offenders is 2.