Science:Math Exam Resources/Courses/MATH102/December 2012/Question B 04

MATH102 December 2012

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Question B 04

A car sales person works on commission and gets paid $2,000 for every car sold. In any month, there is a 0.7 probability of selling exactly one car, a 0.2 probability of selling exactly two cars and a 0.1 probability of selling exactly three cars. Nobody has ever sold more than 3 cars in a month. He gets a bonus of $1000 if he sells three cars in a single month. What is the sales person's expected monthly income?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
This question is asking you to find expected value. To find expected value, you add up possible values (in this case, amount of money made in a month), multiplied by their corresponding probabilities.

Hint 2
The expected value must be between the two extreme values $2,000 and $7,000.

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Since we have three possible outcomes here (selling 1, 2, or 3 cars), we will use a formula for expected value that looks like this: $\displaystyle E[X]=x_{1}p_{1}+x_{2}p_{2}+x_{3}p_{3}$ Where $x_{i}$ is the money earned (or value) of selling i cars in a month and $p_{i}$ is the corresponding probability of selling i cars. Plugging in the appropriate probabilities and amount earned per car (plus the $1000 dollar bonus if 3 cars are sold!), we get: $\displaystyle E[X]=2000*.7+4000*.2+7000*.1$ Simplifying gives: $\displaystyle E[X]=2900$ So the sales person's expected monthly income is $2,900.

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