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There are 10 women and 3 men in Room A. One person is picked [#permalink]

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12 Aug 2010, 03:52

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There are 10 women and 3 men in Room A. One person is picked at random from Room A and moved to room B, where there are already 3 women and 5 men. If a single person is then to be picked from room B, what is the probability that a woman will be picked?

What is the OA? If M picked from room A, room B probability of picking W is 4/9 If W picked from room A, room B probability of picking W is 3/9

Conditional Probability P(W in B and M picked in A) = P(W given M picked in A)*P(M picked in A) = 10/13*4/9 P(W in B and W picked in A) = P(W given W picked in A)*P(W picked in A) = 3/13*3/9

There are 10 women and 3 men in Room A. One person is picked at random from Room A and moved to room B, where there are already 3 women and 5 men. If a single person is then used to be picked from Room B, what is the probability that a woman would be picked.

{Please try solving the problem using the Conditional Probability formula}....Would be very helpful to know how to determine the probability of 2 events when occurring simultaneously}

Using a tree diagram ( see the attachement)

Hence,

WW = \(\frac{10}{13}*\frac{4}{9}=\frac{40}{117}\)

MW = \(\frac{3}{13}*\frac{3}{9}=\frac{9}{117}\)

Finally, the probability that a woman would be picked is \(P= \frac{40}{117} + \frac{9}{117}\)=\frac{49}{117}

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Prob.png [ 9.74 KiB | Viewed 9513 times ]

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Re: There are 10 women and 3 men in Room A. One person is picked [#permalink]

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15 Sep 2013, 21:17

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The probability that a woman is picked from room A is 10/13 the probability that a woman is picked from room B is 4/9. Because we are calculating the probability of picking a woman from room A AND then from room B, we need to multiply these two probabilities: 10/13 x 4/9 = 40/117 The probability that a man is picked from room A is 3/13. If that man is then added to room B, this means that there are 3 women and 6 men in room B. So, the probability that a woman is picked from room B is 3/9. Again, we multiply thse two probabilities: 3/13 x 3/9 = 9/117 To find the total probability that a woman will be picked from room B, we need to take both scenarios into account. In other words, we need to consider the probability of picking a woman and a woman OR a man and a woman. In probabilities, OR means addition. If we add the two probabilities, we get: 40/117 + 9/117 = 49/117 The correct answer is B.

Re: There are 10 women and 3 men in Room A. One person is picked [#permalink]

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01 Jan 2015, 12:08

ARUNPLDb wrote:

The probability that a woman is picked from room A is 10/13 the probability that a woman is picked from room B is 4/9. Because we are calculating the probability of picking a woman from room A AND then from room B, we need to multiply these two probabilities: 10/13 x 4/9 = 40/117 The probability that a man is picked from room A is 3/13. If that man is then added to room B, this means that there are 3 women and 6 men in room B. So, the probability that a woman is picked from room B is 3/9. Again, we multiply thse two probabilities: 3/13 x 3/9 = 9/117 To find the total probability that a woman will be picked from room B, we need to take both scenarios into account. In other words, we need to consider the probability of picking a woman and a woman OR a man and a woman. In probabilities, OR means addition. If we add the two probabilities, we get: 40/117 + 9/117 = 49/117 The correct answer is B.

Why do we need to multiply with the probabilities of woman/man picked from room A. After a person is moved from A to B, we will have either 3 women or 4 women. So why not just add 3/9 + 4/9?? Why to bother about the probability of picking a person from A??

You have to factor in the probability that a man or a woman is transferred from Room A to Room B because THAT outcome affects the probability of the next calculation. While you are correct that there will either be 3 women or 4 women in the room, the probability of one or the other is NOT the same.

Missing that part of the calculation is the equivalent of thinking "there are 3 women and 6 men in a room, so randomly picking 1 person can only lead to 2 results: 1 man or 1 woman. Thus, the odds of picking a woman are 1 in 2." Probability questions on the GMAT are almost always "weighted" - the number of each option affects the probability/calculation, so you have to factor in the "weights."

There are 10 women and 3 men in Room A. One person is picked [#permalink]

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16 Jul 2016, 12:22

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saurabh99 wrote:

ARUNPLDb wrote:

The probability that a woman is picked from room A is 10/13 the probability that a woman is picked from room B is 4/9. Because we are calculating the probability of picking a woman from room A AND then from room B, we need to multiply these two probabilities: 10/13 x 4/9 = 40/117 The probability that a man is picked from room A is 3/13. If that man is then added to room B, this means that there are 3 women and 6 men in room B. So, the probability that a woman is picked from room B is 3/9. Again, we multiply thse two probabilities: 3/13 x 3/9 = 9/117 To find the total probability that a woman will be picked from room B, we need to take both scenarios into account. In other words, we need to consider the probability of picking a woman and a woman OR a man and a woman. In probabilities, OR means addition. If we add the two probabilities, we get: 40/117 + 9/117 = 49/117 The correct answer is B.

Why do we need to multiply with the probabilities of woman/man picked from room A. After a person is moved from A to B, we will have either 3 women or 4 women. So why not just add 3/9 + 4/9?? Why to bother about the probability of picking a person from A??

Thanks, Saurabh

Hi Saurabh,

Picking a member from room B is a dependant event. What is it dependant on ?

As the question reads out " one person is picked from room A AND moved to room B. If a single person is THEN to be picked from B"--> Here FIRST a person is moved THEN picked. So whenever you see such a dependancy , you need to first figure the number of ways of doing the first action.

Whats the first event/action ? Picking and moving a person from room A.

What are our options for event 1 ? Either a man or a woman will be picked.

Hence P(W)= 10/13 or P(M) = 3/13

Now why do we multiply ?

Lets say from point A to B there are 2 ways & from point B to C there are 2 more ways ( No direct route from A to C). How many ways do you have from A to C ?

Total number of ways from A to C= ( # of way from A to B ) * (# of ways from B to C) = 2*2 =4

Coming back to the original question:

Case 1: A woman was picked from room A and a woman was picked from room B

P(W from room A|| W from room B)= (10/13) * (4/9) Case 2: A man was picked from room A and a woman was picked from room B

P(M from room A|| W from room B)= (3/13) * (3/9)

Total probability: case 1 + case 2 ( This is an or case wherein you add the probabilities)

Re: There are 10 women and 3 men in Room A. One person is picked [#permalink]

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25 Jul 2017, 21:29

There are 10 women and 3 men in Room A. One person is picked at random from Room A and moved to room B, where there are already 3 women and 5 men. If a single person is then to be picked from room B, what is the probability that a woman will be picked?

You can think of this as a weighted average problem.

Depending on if you pick a man or a woman from the first room, there are two scenarios with respect to the second room: (1) 4 women and 5 men ( \frac{4}{9} are women), or (2) 3 women and 6 men ( \frac{3}{9} are women).

Now, there are 10 women and 3 men in the first room. So those weights should be used accordingly. Thus...

There are 10 women and 3 men in Room A. One person is picked at random from Room A and moved to room B, where there are already 3 women and 5 men. If a single person is then to be picked from room B, what is the probability that a woman will be picked?

We have two scenarios: when a woman is picked from room A and when a man is picked from room A.

Scenario 1. Let’s start with the woman:

The probability of selecting a woman from room A is 10/13.

If that woman is moved to room B, there are now 4 women and 5 men in room B, and thus, the probability of selecting a woman from room B is 4/9.

For Scenario 1, the overall probability of selecting a woman is 10/13 x 4/9 = 40/117.

Scenario 2. However, if a man is selected from room A: The probability of selecting a man from room A is 3/13.

If that man is moved to room B, there are now 3 women and 6 men in room B, and thus, the probability of selecting a man from room B is 6/9, or ⅔. This means that the probability of selecting a woman from room B is 3/9, or ⅓.

For Scenario 2, the overall probability of selecting a woman is 3/13 x 3/9 = 9/117.

So, the probability of selecting a woman from room B is 40/117 + 9/117 = 49/117.

Answer: B
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