Darcy, Gina, Ray and Susan will be the only participants at a meeting.

Question

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%

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%

chriswil2005 · Accepted Answer

I got B. This is how I got it. The prob that Darcy will sit in a soft chair=prob(arriving first)+prob(arriving second)+(0.5)prob(arriving third). If she arrives third, she will be arriving with Ray and since there will be one soft chair remaining they flip a coin for it. That is why I halfed the last term in the prob. Since we know that the prob of Gina or Susan to arrive first is 1/3 each, then the prob of Darcy and Ray to arrive first is also 1/3. From this knowledge we, can use the info in the stemn to deduce that the prob of Darcy and Ray to arrive second or last is also 1/3 for each instance. Therefore
prob of getting soft chair is 1/3+1/3+(0.5)(1/3)=5/6

The prob of Gina sitting in the soft chair, can happen only if she arrives first or second, so the prob=1-prob(arriving last)=1-1/3=2/3.

From here, use  /2/3

This equals 25% (b). I know this was very confusing, and I may not even be correct. What is the OA? If I am correct, I will try an explain it more if needed.

gmatchile · Answer

We will do:

Darcy = D
Ray = R
Susan = S
Gina = G

Keep in mind that follows:

1 Darcy and Ray arrive together at the location of the chairs.

2. the probability that Susan arrives first is 1/3 and the probability that Gina arrives first is also 1/3.

3. the probability that Gina comes last is 1/3 and the probability that Susan comes last also is 1/3.

1.-deduce that DR, come as a unit to the location of the chairs, so we will have three candidates (not four) arriving at the location of the chairs, these are: DR, S and G.

2.-given that there are three candidates for the position of the chairs, it follows that the probability of arriving first DR is 1/3.

3.-given that there are three candidates for the position of the chairs, it follows that the probability of arriving last DR is 1/3

We can also deduce from all of the above information to: as each of the three: DR, G or S, may become second and that between the probability that arrive in the first or the last place each of the three add 2/3, we conclude that each of the three has a probability of 1/3 in second place.

Thus we have arrival location of chairs

DR probability:

For 1st place = 1/3
For 2nd place = 1/3
For 3rd place = 1/3

Probability of S:

For 1st place = 1/3
For 2nd place = 1/3
For 3rd place = 1/3

Probability of G:

For 1st place = 1/3
For 2nd place = 1/3
For 3rd place = 1/3

So:

Location of D to access a soft Chair

If DR comes in first or second place both as R D they will access a soft Chair, this means that D in this situation has 1/3 + 1/3 to a soft Chair.

If DR comes third (last), D has a probability of (1/2) x (1/3) access to a soft Chair.

Then in all cases D has a probability of 1/3 + 1/3 + (1/2) x (1/3) = 5/6 access to a soft Chair.

Location of G to access a soft Chair

To G enter a soft Chair, you can only get in first or second place, if it reaches third we have addressed three soft chairs DR two soft chairs and S a soft Chair.
The probability that G arrives in first or second place is 1/3 + 1/3 = 2/3 = 4/6

Finally the highest percentage who D on G accesses a soft Chair is:

100 x (5/6 - 4/6) /(4/6) = 25% correct response b)

hurtado Claudio gmat gre sat math preparation

automan · Answer

These are all the possible combinations:

S takes a hard chair
DGRS--1/6
GRDS--1/6

D takes a hard chair
SGRD--1/6
GRSD--1/6

R takes a hard chair
SDGR--1/12
DSGR--1/12

G takes a hard chair
SDRG--1/12
DSRG--1/12

Therefore, the probability that Darcy will sit in a hard chair is 1/3 and the probability that Darcy will sit in a soft chair is 2/3

And the probability that Gina will sit in a hard chair is 1/6 and the probability that Gina will sit in a soft chair is 5/6.

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? . This is a ticky question to answer as the first probablity is lower than the second one.

I would answer 0.

If the question were asking the opposite, ie by what percent the probability that Gina will sit in a soft chair greater than the probability that dary will sit in a soft chair is, the answer would be 25%.

GMATT73 · Answer

The prob of Gina sitting in the soft chair, can happen only if she arrives first or second, so the prob=1-prob(arriving last)=1-1/3=2/3.

From here, use [(5/6)-(2/3)]/2/3

These were the two crucial steps that stumped me. Thanks for putting it all together Chris :-D

ywilfred · Answer

The probablity that Gina arrives third and does not get a soft chair really depends on whether Darcy and Ray want to take up a soft chair at all Doesnt it? 

If Susan comes first, and picks the hard chair, then it doesn't really matter if Gina comes third or even last since she will always gets soft chair in this instance. To me, this question has a lot of assumptions.

chriswil2005 · Answer

I assume that if there is an empty soft chair, you must take it. The stem states, "Upon arriving at the meeting, each of the participants will select a soft chair, if one is available". 

GMATT, what is the OA?

ywilfred · Answer

I believe that your last step should just be 1/6*100% = 16.66%. We're asking percentage differecne and since we already found the difference (by subtracting the probability), then we should just conver it to a percentage change.

GMATT73 · Answer

OA is B. This is from the Kaplan course book. OE is a full page in length, but Chris pull it together in a nutshell.

nawaf52 · Answer

Hello Guys
I Understood the process of getting 2/3 and 4/5 . But I do not understand why we divided the difference between them by 2/3. Could someone please explain that for me?

albo94 · Answer

Because we have to find how much higher is D's probability (of getting a soft chair) than G's one. Not the absolute difference between the two probablities. Is it clear ?

ShashankDave · Answer

IanStewart   VeritasPrepKarishma   Bunuel

Experts, I need help with some questions regarding this.

1.Are we keeping track of the order in which they arrive AND take the chairs? or just the order in which they arrive?

2.How do we create the complete sample space for this question?

3.What is the probability of the event - Gina arrives first, Susan arrives second, and Darcy and Ray arrive last?

4. The probability that some one will arrive first, somebody second, and somebody third are all individually 1 (1/3 + 1/3 + 1/3). The probability that all of them will arrive is also 1, as we are sure all of them will certainly arrive at some point. From the provided information, how do I calculate this probability? in other words, how to make the probabilities sum up to 1?

This will solve many problems for me and make me much better at probability. Please help.

AdityaVerma21 · Answer

well Explained , but you have exchanged G and D.

Regor60 · Answer

The 3 groups each have the same probability of showing up in any position, 1/3.

Since there are 3 groups, there are 3!=6 ways all 3 show up and all 6 are equally likely.

D is guaranteed a seat if she and R show up first or second.

There are 2 ways for D&R to show up first, since G and S each can be 2nd or 3rd.

There are 2 ways for D&R to show up 2nd, since G&S each can be 1st or 3rd.

Finally, there are 2 ways D&R can show up last by the same logic, but only a 50% chance that D gets a seat in each, so 1 way equivalent.

Total ways for D=5

G is guaranteed a seat if she shows up 1st or 2nd. These each can happen 2 ways since D&R and S can swap positions in both, so a total of

4 ways for G. So the percentage difference is

100*(5-4)/4= 25%

Posted from my mobile device

mitul19 · Answer

i have a doubt.: I did like this: Prob. of darcy arriving first = Prob.(Gina not arriving First) * Prob.(Susan not arriving first) = (1-1/3)(1-1/3) = 2/3*2/3 = 4/9 Can someone pls explain why this isn't correct? Thanks in advance.

Regor60 · Answer

Consider that both Susan and Gina have a 1/3 chance of showing up first, and that SOMEONE will with a probability of 1 show up first, that must mean that there is 1-1/3-1/3 = 1/3 chance for the remaining 2 people, Darcy and Ray since they show up together, to show up first.

If that's not obvious, let's drill down into your 2/3*2/3 calculation.

That's saying:

(1/3+1/3)*(1/3+1/3) properly captures the probability of neither Gina or Susan being first, with the first 1/3 in each above being the probability of arriving 2nd and the 2nd 1/3 being the probability of arriving 3rd.

So multiplying the first terms above is saying that there is 1/3*1/3=1/9 chance of BOTH arriving 2nd.

We know that can't happen.

Likewise multiplying both 2nd terms together is saying there's a 1/9 chance of BOTH arriving 3rd, which we also know can't happen.

So the 4/9 has to be reduced by 2*1/9= 2/9.

But we're not finished.

Multiplying the first term above against the second term in the other parenthesis expression is saying there's a 1/9 chance of Gina arriving 2nd and Susan 3rd. Remember, we're already assuming Darcy/Ray arrive first

But that's not true. If Gina arrives 2nd there's a 100% chance Susan arrives 3rd.

So the probability of this is UNDERSTATED by 1/3-1/9=2/9.

This same understatement occurs when Susan arrives 2nd and Gina 3rd.

So we need to add 2/9+2/9=4/9
to our adjusted 2/9 we previously calculated.

This equals 6/9 or 2/3, which is the accurate number to be subtracted from 1 to determine Darcy/Ray chance of arriving 1st, and that equals

1/3

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

Darcy, Gina, Ray and Susan will be the only participants at a meeting.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Darcy, Gina, Ray and Susan will be the only participants at a meeting.

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards