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Hi Clyde,

I'm glad to see you're good at probability. Would you help me with this question I've encountered while doing a CMA review course?

"A car rental agency has 2 locations. 60% of the cars rented from Winter City are returned to the Winter City office while 40% are returned to San Soleil.

Further, approximately 20% of those rented from San Soleil are returned to Winter City and 80% are returned to San Soleil.

What is the joint probability that a car rented from Winter City at the beginning of week 1 will be returned to Winter City at the end of week 2? Assume that one week represents one period."

The figuring I've done thus far has led me to make a tree diagram where, on the right-hand stem, there is a 60% chance of the car being returned to WC in Wk 1. On the left-hand stem, there is a 40% chance it won't be returned to WC in Wk 1. Then, attached to that left-hand stem, there is a 60% chance of it being returned to WC in Wk 2.

Finally, I came out with this: P = 40% x 60% + 60% = 84% chance that the car, rented at the beginning of Wk 1, would be returned to WC at the end of Wk2.

This is, as the answer choices indicate, incorrect. Would you lend me your help?

Thanks,

Jeff

Answer
Tree
Tree  
You have four possible outcomes.

Let "W" and "S" represent where the car is, and times t=0, 1, and 2 be the times in question (zero is at the beginning, 1 is after one week, 2 is after 2 weeks).

t = 0, W

t = 1, WW or WS

t = 2, WWW, WWS, WSW, WSS

Those are the four possible places the car will "check in" so the total probability is:

P(WWW) + P(WWS) + P(WSW) + P(WSS) = 1

since those are the only possible outcomes.

However, you are interested in:

P(WWW) + P(WSW) = ?

Those are the two outcomes where it starts in W and ends up in W. Here, you can compute each of these directly.


For WWW, it has to go from W to W, and then again from W to W. That is:

P(WWW) = 60% * 60% = 0.6 * 0.6 = 0.36

For WSW, it goes from W to S, and then from S to W:

P(WSW) = 40% * 20% = 0.4 * 0.2 = 0.08

So the total is only P(WWW)+P(SWS) = 0.44 = 44%



To verify that this is true, apply the same logic:

P(WSS) = 40% * 80% = 32%

P(WWS) = 60% * 40% = 24%

P(WSS) + P(WWS) = 56%

That must be right, since it is the complementary probability to our answer of 44%.



This can be visualized using a probability tree (see attached image).



I assume 44% is one of your choices, yes?

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Clyde Oliver

Expertise

I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

Experience

I am a PhD educated mathematician working in research at a major university.

Organizations
AMS

Publications
Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

Education/Credentials
BA mathematics & physics, PhD mathematics from a top 20 US school.

Awards and Honors
Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

Past/Present Clients
In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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