This exercise utilizes the binomial distribution. According to the website nationalbikeregistry, at the UC Berkeley campus, only 3% of stolen bicycles are returned to their owners. Suppose 235 bikes are stolen in a recent year.

(a) What is the fixed number of trials, n? What is the success probability, p?
(b) Find the probability that exactly 12 of the stolen bicycles are returned to their owners.
(c) Find the probability that at most 12 of the stolen bicycles are returned to their owners.
(d) Calculate the mean number of bicycles that will be returned to their owners in a year when 235 are stolen. Use proper units and interpret this mean.
(e) Calculate the standard deviation number of bicycles that will be returned to their owners in a year when 235 are stolen. Use proper units.

a) n = 235

p = 0.03

b) P(X=12) = 0.02655

c) P(X ≤ 12) = 0.9737

d) Mean = μ = 7.05 stolen bicycles would be returned.

This means that for the 235 stolen bicycles in the year under review, 7.05 of them are expected to be returned.

e) Standard deviation = σ = 2.62 returned bicycles.

Step-by-step explanation:

Binomial Distribution problems are solved with the Binomial distribution function.

P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ

n = total number of sample spaces = number of stolen bicycles = 235

x = Number of successes required = variable; depending on the question

p = probability of success = Probability of a bike being returned = 3% = 0.03

q = probability of failure = Probability of a bike not being returned = 1 - 0.03 = 0.97

a) The fixed number of trials = number of stolen bikes in a year under review = n = 235

p = success probability = Probability of a bike being returned = 3% = 0.03

b) The probability that exactly 12 of the stolen bicycles are returned to their owners

P(X=12)

n = 235, p = 0.03, q = 0.97, x = 12

P(X=12) = ²³⁵C₁₂ (0.03)¹² (0.97)²³⁵⁻¹²

P(X=12) = 0.0265471215 = 0.02655

c) The probability that at most 12 of the stolen bicycles are returned to their owners. P(X ≤ 12)

P(X ≤ 12) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7) + P(X=8) + P(X=9) + P(X=10) + P(X=11) + P(X=12)

Solving each of these and summing them all up or using the Binomial distribution calculator,

P(X ≤ 12) = 0.97368510024 = 0.9737

d) Mean for Binomial distribution is given as

Mean = np = (235)(0.03) = 7.05 stolen bicycles would be returned.

This is the number of stolen bicycles that would be returned.

This means that for the 235 stolen bicycles in the year under review, 7.05 of them are expected to be returned.

e) Standard deviation for Binomial distribution is given as

σ = √[np(1-p)] = √(235×0.03×0.97) = 2.615 ≈ 2.62 returned bicycles.

Hope this Helps