Question

A computer company shipped two new computers to a customer. These two computers were randomly selected from the 15 computers in stock. Unfortunately, the inventory clerk by mistake mixed up new computers with refurbished computers. As a result, the 15 computers in stock consisted of 11 new computers and 4 refurbished computers. If the customer received one refurbished computer, the company will incur a shipping and handling expense of $100 to replace that computer with a new computer. However, if both computers were refurbished, the customer would cancel the order and the company will incur a total loss of $1,000. If the customer received both computers as new, then there is no extra cost involved (i.e., zero loss). Find the expected value and standard deviation of the computer company's loss.

Answer 1

Expected value of company loss = $99.01

Standard deviation = $226.91

Explanation:

We first obtain the probability mass function of the company's losses based on the chances of the possible various number of refurbished computers in the customers order.

There are 15 total computers in stock.

There are 4 refurbished computers in stock.

There are 11 new computers in stock

The customer orders 2 computers. If there are no refurbished computers in the order, there are no losses on the company's part.

Probability of no refurbished computers in the order = (11/15) × (10/14) = 0.5238

The customer orders 2 computers. If there is only 1 refurbished computer in the order, there is a loss of $100 on the company's part.

Probability of 1 refurbished computers in the order = [(11/15) × (4/14)] + [(4/15) × (11/14)]

= 0.4191

The customer orders 2 computers. If there are 2 refurbished computer in the order, there is a loss of $1000 on the company's part.

Probability of 2 refurbished computers in the order = [(4/15) × (3/14)] = 0.0571

So, the Probabilty function of random variable X which represents the possible losses that the company can take on is given as

X | P(X)

0 | 0.5238

100 | 0.4191

1000 | 0.0571

Expected value of company loss is given as

E(X) = Σ xᵢpᵢ

xᵢ = each variable or sample space

pᵢ = probability of each variable

E(X) = (0 × 0.5238) + (100 × 0.4191) + (1000 × 0.0571) = $99.01

Standard deviation is obtained as the square root of variance.

Variance = Var(X) = Σx²p − μ²

where μ = E(X) = 99.01

Σx²p = (0² × 0.5238) + (100² × 0.4191) + (1000² × 0.0571) = 0 + 4191 + 57,100 = 61,291

Var(X) = Σx²p − μ²

Var(X) = 61291 − 99.01² = 51,488.0199

Standard deviation = √(51,488.0199) = $226.91

Hope this Helps!!!