Question

On Tuesday, Sara’s Produce is expecting to receive Package A containing $2,000 worth of food. Based on past experience with the delivery service, the owner estimates that this package has a chance of 10% being lost in shipment.

On Wednesday, Sara’s Produce expects Package B to be delivered. Package B contains $1,000 worth of food. This package has a 4% chance of being lost in shipment.

1. Construct [in table form] the probability distribution for the total dollar amount of losses for package A and B. [4 points]

In the table, make sure you specify:

- The possible outcomes for Sara’s total dollar amount of losses for packages A and B. Please note that this asks about the total dollar amount of losses, not the number of losses.

- For each dollar amount of losses, describe under what circumstances it would occur. In other words, what event(s) must happen for each dollar amount of losses to occur?

- For each of the possible outcomes, you identify in part [a], derive the probability of the outcome occurring.

2. Calculate the expected value of the total dollar amount of losses. [1 point]
3. The owner has calculated the variance for the total dollar amount of losses to be 398,400. Since you want to be sure you are using correct numbers in your evaluation, prove that the owner calculated the correct variance for the total dollar amount of losses. Show all work! [2 points]

Answer 1

Answer:

a. Let X be the amount of losses for package A and B

P(X = 0) = P(Package A not lost, Package B not lost)

= (1 - 0.1) * (1 - 0.04)

= (0.9) * (0.96)

= 0.864

P(X = 2000) = P(Package A lost, Package B not lost)

= 0.1 * (1 - 0.04)

= 0.1 * 0.96

= 0.096

P(X = 1000) = P(Package A not lost, Package B lost)

= (1 - 0.1) * 0.04

= 0.9 * 0.04

= 0.036

P(X = 3000) = P(Package A lost, Package B lost)

= 0.1 * 0.04

= 0.004

The probability distribution for total dollar amount of losses for package A and B is,

X	P(X)
0	0.864
1000	0.036
2000	0.096
3000	0.004

b.
Expected value of total dollar amount of losses,

E(X) = 0 * 0.864 + 1000 * 0.036 + 2000 * 0.096 + 3000 * 0.004

= 0 + 36 + 192 + 12
= $240
c.
E( X2) = 02 * 0.864 + 10002 * 0.036 +20002 * 0.096 + 30002 * 0.004

= 0 + 36000 + 384000 + 36000

= 456000
Variance for the total dollar amount of losses = E(X2) - [E(X)]2

= 456000 - 2402

= 398400

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