Answer to Question #167545 in Statistics and Probability for sami

Question

A particular type of smart phone comes in a regular-size version and a “plus-size” version. 60% of all customers prefer the “plus-size” version.

(a) Among 10 randomly selected customers who want this type of smart phone, what is the probability that at least eight want the “plus-size” version?

(b) Among 10 randomly selected customers, what is the probability that the number of customers who want the “plus-size” version is within 1 standard deviation of the mean value? [Hint: what probability distribution should be used, and what are the associated mean and standard deviation?]

(c) The store currently has 7 smart phones of each version. What is the probability that all of the next 10 customers who want this smart phone can get the version they prefer from the current stock? [Hint: what is the value range for # of customers who want the “plus-size” given the restricted stock?]

(d) The store is retailing the regular-size smart phone for $400 and the “plus-size” one for $600, and the store has 20 smart phones for each version in stock. What is the expected revenue from the purchases of the next 3 customers?

Answer 1

1. We know that the probability of choosing a “plus-size” version by a person is 0.6. At least 8 means 8 or 9 or 10 people in our case and we need to summarize these three probabilities. If there are 8 people, the probability is 0.6⁸ * 0.4² * C₁₀⁸, where C₁₀⁸ = 10! / (8! * (10-8)!), 10! = 10*9*8*...*2*1 means the amount of variants to choose 8 of 10 without importance of order. Hence we need to calculate the probability 0.6⁸ * 0.4² * C₁₀⁸ + 0.6⁹ * 0.4¹ * C₁₀⁹ + 0.6¹⁰ * 0.4⁰ * C₁₀¹⁰ = 0.121 + 0.042 + 0.006 = 0.17...

Let's consider binomial distribution. The standard deviation is

$$\sqrt{n*p*(1-p)}$$

1. = 1.54. Hence the interval is (6 - 1.54, 6 + 1.54) = (4.46, 7.54) , but the numbers must be integer, so the interval is (5 , 7). The probability is 0.6⁵ * 0.4⁵ * C₁₀⁵ + 0.6⁶ * 0.4⁴ * C₁₀⁶ + 0.6⁷ * 0.4³ * C₁₀⁷ = 0.20 + 0.25 + 0.21 = 0.66
2. This case is the same if we want to find the probability of event when the number of customers preferring “plus-size” less than 8 and more than 2. It's equal to 1 - 0.6⁸ * 0.4² * C₁₀⁸ - 0.6⁹ * 0.4¹ * C₁₀⁹ - 0.6¹⁰ * 0.4⁰ * C₁₀¹⁰ - 0.6¹ * 0.4⁹ * C₁₀¹ + 0.6² * 0.4⁸ * C₁₀² = 1 - 0.17 - 0.01 = 0.82
3. Expected value of one purchase is (600+400) / 2 = 500. Hence he expected revenue from the purchases of the next 3 customers is 500 * 3 = 1500