Advanced Probability - Who Gets a Seat?

Question

Not only is it long, but it is complicated. Yet, despite knowing the answer, I have been stumped more than once. I welcome other unique approaches from various members.
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Darcy, Gina, Ray and Susan will be the only participants at a meeting. There will be three soft chairs in the room where the meeting will be held, and one hard chair. No one can bring more chairs into the room. Darcy and Ray will arrive simultaneously, but Gina and Susan will arrive individually. The probability that Gina will arrive first is 1/3, and the probability that Susan will arrive first is 1/3. The probability that Gina will arrive last is 1/3, and the probability that Susan will arrive last is 1/3. Upon arriving at the meeting, each of the participants will select a soft chair, if one is available. If Darcy and Ray arrive and see only one unoccupied soft chair, they will flip a fair coin to determine who will sit in that chair. By what percent is the probability that Darcy will sit in a soft chair greater than the probability that Gina will sit in a soft chair?

A 50%
B 25%
C 16 2/3%
D 12 1/2%
E 0%

Source: Kaplan GMAT

wawatan · Answer

wow long problem.

Ian Stewart · Answer

beeparoo wrote:Darcy, Gina, Ray and Susan will be the only participants at a meeting. There will be three soft chairs in the room where the meeting will be held, and one hard chair. No one can bring more chairs into the room. Darcy and Ray will arrive simultaneously, but Gina and Susan will arrive individually. The probability that Gina will arrive first is 1/3, and the probability that Susan will arrive first is 1/3. The probability that Gina will arrive last is 1/3, and the probability that Susan will arrive last is 1/3. Upon arriving at the meeting, each of the participants will select a soft chair, if one is available. If Darcy and Ray arrive and see only one unoccupied soft chair, they will flip a fair coin to determine who will sit in that chair. By what percent is the probability that Darcy will sit in a soft chair greater than the probability that Gina will sit in a soft chair?

The question is overly long, certainly longer than most real GMAT questions. The question is just trying to say that Gina, Susan and Darcy/Ray show up in random order- each has a 1/3 chance of showing up first, second, or last.

What's the probability that Gina will sit in a soft chair? If she shows up first or second, there will certainly be a soft chair available. If she shows up last, there won't. So 2/3.

What's the probability that Darcy will sit in a soft chair? If Darcy and Ray show up first or second, Darcy will get a soft chair: 2/3. If Darcy and Ray show up last, they flip a coin: 1/3 * 1/2 = 1/6. So the probability is 2/3 + 1/6 = 5/6 that Darcy gets a soft chair.

Finally, 5/6 is what percent greater than 2/3? (5/6 - 2/3)/(2/3) = 25%

For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

ianstewartgmat.com

beeparoo · Answer

Wow, Ian. Your confidance throughout your solution is unwavering. I hope to build that level of "but of course!" attitude on GMAT.

Anyway, I was unsure of my footing during the calculation of Darcy's probability and especially Gina's, thinking that I should treat two instances separately:

For calculation of Gina as second arrival, consider the possibility that
1) Susan enters first
or
2) Darcy/Gina enter first

Thoughtfully reviewing your response, I see, in retrospect, how they don't need to be quantified separately. But hindsight is 20/20 and clearly, I need to polish my probability skills.

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