Question

Seventy percent of the bicycles sold by a certain store are mountain bikes. Among 100 randomly selected bike purchases, what is the approximate probability that a. At most 75 are mountain bikes? b. Between 60 and 75 (inclusive) are mountain bikes? c. More than 80 are mountain bikes? d. At most 30 are not mountain bikes?

Short answer

a. The probability that at most 75 bikes are mountain bikes is approximately 0.862. \b. The probability that between 60 and 75 bikes are mountain bikes is approximately 0.848. \c. The probability that more than 80 bikes are mountain bikes is approximately 0.015. \d. The probability that at most 30 bikes are not mountain bikes is 0.5.

Step-by-step solution

Calculate Mean and Standard Deviation (SD)

The mean (μ) of the distribution is given by np (n=number of trials, p=probability of success). So, μ = np = 100*0.7 = 70. \The standard deviation (σ) for a binomial distribution is given by √(np(1 - p)) = √(100*0.7*(1 - 0.7)) = √21 ≈ 4.58.

Calculate Z-scores and Finding the probabilities

The Z-score is calculated as Z = (X - μ)/σ. The probabilities can be found by using standard normal distribution table. \ a. 'At most 75 are mountain bikes' means X ≤ 75. So, Z = (75 - 70)/4.58 ≈ 1.09. The probability that Z ≤ 1.09 is 0.8621. \ b. ‘Between 60 and 75 (inclusive) are mountain bikes' means 60 ≤ X ≤ 75. So we calculate two Z-scores: Z1 = (60 - 70)/4.58 ≈ -2.18 and Z2 = (75 - 70)/4.58 ≈ 1.09. The probability that -2.18 ≤ Z ≤ 1.09 is 0.8621 - 0.0146 = 0.8475. \ c. 'More than 80 are mountain bikes' means X > 80. So, Z = (80 - 70)/4.58 ≈ 2.18. The probability that Z > 2.18 is 1 - 0.9854 = 0.0146. \ d. 'At most 30 are not mountain bikes' is the same as 'At least 70 are mountain bikes', means X ≥ 70. So, Z = (70 - 70)/4.58 = 0. The probability that Z ≥ 0 is 0.5.

Wrapping up the Results

a. The probability that at most 75 bikes are mountain bikes is 0.8621. \b. The probability that between 60 and 75 bikes are mountain bikes is 0.8475. \c. The probability that more than 80 bikes are mountain bikes is 0.0146. \d. The probability that at most 30 bikes are not mountain bikes is 0.5. Thus, all the probabilities for different scenarios have been found.