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

529 and 531

step-by-step explanation:

the most boring ever written

step-by-step explanation:

i am writing something. yes, i plan to make it the most boring thing ever written. i go to the store. a car is parked. many cars are parked or moving. some are blue. some are tan. they have windows. in the store, there are items for sale. these include such things as soap, detergent, magazines, and lettuce. you can enhance your life with these products. soap can be used for bathing, be it in a bathtub or in a shower. apply the soap to your body and rinse. detergent is used to wash clothes. place your dirty clothes into a washing machine and add some detergent as directed on the box. select the appropriate settings on your washing machine and you should be ready to begin. magazines are stapled reading material made with glossy paper, and they cover a wide variety of topics, ranging from news and politics to business and stock market information. some magazines are concerned with more recreational topics, like sports card collecting or different kinds of hairstyles. lettuce is a vegetable.