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Worked solution

Find the expected commission

A salesperson for a medical device company has two appointments. He believes the first has a 70% chance of a deal paying 1,000 dollars commission, and the second a 40% chance of a deal paying 1,500 dollars. Assume the appointment results are independent. What is his expected commission based on these beliefs?

CJC · Lecture · Example 9

Build the complete commission distribution

Let XX be the total commission in dollars. The four cases are no deals, first only, second only and both. Independence gives the Teacher-completed probabilities and contributions:

Commission xx0100015002500
P(X=x)P(X=x)(0.3)(0.6)=0.18(0.3)(0.6)=0.18(0.7)(1−0.4)=0.42(0.7)(1-0.4)=0.42(1−0.7)(0.4)=0.12(1-0.7)(0.4)=0.12(0.7)(0.4)=0.28(0.7)(0.4)=0.28
xP(X=x)xP(X=x)0420180700
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Add the weighted amounts

E(X)=0+420+180+700=1300E(X)=0+420+180+700=1300. Hence the expected commission is 1,300 dollars. It is a probability-weighted average based on the stated beliefs, not a promised payment or a fifth possible commission amount.

How did this part go?

Common mistakes

Keep the two appointment probabilities separate. Independence is given for the source’s joint probability products; the expected total is a long-run average, not the most likely outcome.

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