Problem #1 (16 points) Part a) You are a a salesperson for a company specializin

Question

Problem #1 (16 points) Part a) You are a a salesperson for a company specializing in fine chocolates. Let X be the differernt types of chocolate your company sells. The second row gives you the percentage of each type of chocolate you sell in a typical year. You’ll also note that two of the cells are missing a value. A pound of Teuscher truffles costs $24, Vosges is $27, Scharffen is $22 and Valrhona is $19. If you anticipate selling 1400 pounds of chocolate this coming year, what is the expected gross income that you will make for the company? X Teuscher Vosges Scharffen Valrhona 0.29 0.41

Part b) (problem #2): A door-to-door salesman selling pens. He sells a fancy 'Mont Bic' pen that makes him $20. He also sells a more basic pen 'Fort Ticonderoga' that makes him $6. He sells that 'Mont Bic' in 1/40 houses, the 'Fort Ticonderoga' in 5/40 houses. He makes no sale in the remaining 34/40 houses. If he goes to 200 houses over the course of a weekend, how much can he hope to make based on his probabilities up to this point?

Problem #3 (12 points) Suppose a certain department is considered to be 'high cost' if their average salary exceeds $150,000 or more. Now suppose that a department is considered 'high volume' if its sales exceeds $1.5 million per quarter. Select a department at random, and let A be the event that the department is 'high cost', and B the department is 'high volume'. According to your records, P('A') = 0.21, P('B') = 0.34, and the probability that both are true, i.e. P(A and B) = 0.068. What is the probability that a randomly selected department is either high cost or high volume?

Problem #4 (12 points) At the olympic games, a group of 20 athletes are chosen at random. 8 of them are wrestlers, 4 are swimmers, and 8 are softball players. If you were to pick three athletes at random, what is the probability that the first one was a wrestler, and the second one was also a wrestler?

Explanation / Answer

1) Since two of the probabilities are missing, we can't calculate the exact number for expected gross income but we can calculate the range of gross incomes.

Remaining probability which needs to be distributed among Scharffen and Valrhona = 1 - (0.41 + 0.29) = 0.3

So the minimum expected gross income would be when the entire 0.3 would be Valrhona.

Minimum expected gross income per chocolate sold = 24(0.29) + 27(0.41) + 19(0.3) = $23.73

Hence,

For 1400 chocolates, minimum expected gross income = 1400($23.73) = $33222

Similarly,

Maximum expected gross income would be when the entire 0.3 would be Scharffen.

Minimum expected gross income per chocolate sold = 24(0.29) + 27(0.41) + 22(0.3) = $24.63

Hence,

For 1400 chocolates, minimum expected gross income = 1400($24.63) = $34482

Hence,

The expected gross income would range between $33222 and $34482.

2) Probability distribution will be:

Expected sell per house = 20(1/40) + 6(5/40) + 0(34/40) = $1.25

Hence,

If he visit 200 houses, the expected sell = 200($1.25) = $250

3) P(High cost or high volume)

= P(High cost) + P(High Volume) - P(High cost and Volume)

= 0.21 + 0.34 - 0.068

= 0.482

4) P(First wrestler and second wrestler)

= P(First wrestler) * P(Second wrestler | First one chosen is wrestler)

= (8/20) * (7/19)

= 14/95

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