A salesman has scheduled two appointments to sell encyclopedias. His first appoi

Question

A salesman has scheduled two appointments to sell encyclopedias. His first appointment will lead to a sale with probability .3, and his second will lead independently to a sale with probability .6. Any sale made is equally likely to be either for the deluxe model, which costs $1000, or the standard model, which costs $500.

A.) Determine the probability mass function of X, the total dollar value of all sales.

B.) What is the expected value of this random variable?

Please show work! Please use tables. Thank you

Explanation / Answer

(A) Pr(First appointment will be a success) = 0.30

Pr(Second appointment will be success) = 0.60

Pr(Deluxe Model) = Pr(Standard model) = 0.50

so there are four possibilities.

1. Both appoitments will be a failure

Total dollar value of the sales X = 0

Probability of that hapening = (1 - 0.3) * (1 - 0.6) = 0.28

2. He will be able to sell one Deluxe model in one of the two appointment

so X = $ 1000

Pr(X = $ 1000) = Pr(First appointment Success) * Pr(Deluxe) * Pr(Second APpointment failure) + Pr(first appointment failure) * Pr(Deluxe) * Pr(Second Appointment success)

= 0.3 * 0.5 * 0.4 + 0.7 * 0.5 * 0.6 = 0.27

3. He will be able to sell two deluxe model in both the appointment

Here total earnings = $ 2000

Pr(X = $ 2000) = Pr(First appointment Success) * Pr(Deluxe) * Pr(Second Appointment success) * Pr(Deluxe)

= 0.3 * 0.5 * 0.6 * 0.5= 0.045

4. He will be able to sell one Standard model in one of the two appointment

so X = $ 500

Pr(X = $ 500) = Pr(First appointment Success) * Pr(standard) * Pr(Second APpointment failure) + Pr(first appointment failure) * Pr(standard) * Pr(Second Appointment success)

= 0.3 * 0.4 * 0.5 + 0.7 * 0.6 *0.5 = 0.27

5. He will be able to sell two stanadrd model in both the appointment

Here total earnings = $ 1000

Pr(X = $ 1000) = Pr(First appointment Success) * Pr(standard) * Pr(Second Appointment success) * Pr(stanadrd)

= 0.3 * 0.5 * 0.6 * 0.5 = 0.045

6. WHen he will be able to sell one standard model in one of the appointment and the deluxe model in other one.

Total earning = $1500

Pr(X = $ 1500) = Pr(First success) * Pr(Deluxe) * Pr(Second success) * Pr(standard) + Pr(first success) * Pr(Standard) * Pr(second success) * Pr(deluxe)

= 0.3 * 0..5 * 0.6 * 0.5 + 0.3 * 0.5 * 0.6 * 0.5 = 0.09

so as we can see that X can take Value of $0, $500, $ 1000, $1500 and $ 2000

so fx(x) = 0.28 ; X = 0

= 0.27; X = $ 500

= (0.27 + 0.045) = 0.315 ; X = $ 1000

= 0.09; X = $ 1500

= 0.045; X = $2000

(B) Expecte value of X

E(X) = 0 * 0.28 + 0.27 * 500 + 0.3125 * 1000 + 0.09 * 1500 + 0.045 * 2000 = $672.5

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