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Intern  Status: Desperate to breach the 700 barrier
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There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 69% (02:23) correct 31% (01:59) wrong based on 270 sessions

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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?

(A) 13/21
(B) 49/117
(C) 40/117
(D) 15/52
(E) 5/18

Originally posted by HarishV on 12 Aug 2010, 04:52.
Last edited by Bunuel on 16 Sep 2013, 04:02, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Probability Question  [#permalink]

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Ans: Probability of 49/117

The way I approached it is (probability of Woman from A * probability of woman from B) +( probability of man from A * probability of woman from B)

so its 10/13 * 4/9 + 3/13+3/9

Solving we get 49/117.

Right?

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Re: Probability Question  [#permalink]

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1
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

Sum 49/117
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Re: Probability Question  [#permalink]

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HarishV wrote:
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}
Attachments Prob.png [ 9.74 KiB | Viewed 15324 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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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.
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Re: There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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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
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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Hi saurabh99,

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."

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

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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)

= (40/117) + (1/13)
= 49/117

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

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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?

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...

$$(10*\frac{4}{9} + 3*\frac{3}{9})$$ /(13) = \frac{49}{117}
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Re: There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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HarishV wrote:
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?

(A) 13/21
(B) 49/117
(C) 40/117
(D) 15/52
(E) 5/18

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.

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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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Hi All,

This question combines the concepts of Weighted Averages and Probability. To stay organized, you might find it best to break the probability down into 'pieces':

Room A has 10 women and 3 men in it, so when you move one of those people to Room B, there's a 10/13 probability of moving a woman and 3/13 probability of moving a man. We have to keep track of both possibilities.

When you add that 1 person to the second room, you'll raise the total number of people there to 9 (but you might be adding a woman or a man, so you have to adjust your math accordingly - there would either be 3 women or 4 women). We're asked for the probability that a woman is then picked from Room B.

IF...
A woman is moved to Room B.... (10/13)(4/9) = 40/117
A man is moved to Room B... (3/13)(3/9) = 9/117

Total probability = 49/117

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

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HarishV wrote:
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?

(A) 13/21
(B) 49/117
(C) 40/117
(D) 15/52
(E) 5/18

10W, 3M - Room A
3W, 5M - Room B

1st Case:

Suppose, woman is picked from Room A - Probability of picking Woman from Room $$A = \frac{10}{13}$$
Now, we have 4W & 5M. So, Probability of picking woman in room $$B = \frac{4}{9}$$

2nd case:

Suppose, a Man is picked from Room A. Probability of picking the Man from Room $$A = \frac{3}{13}.$$
Now, we have 3W & 6M in Room B. So probability of picking Woman from Room $$B = \frac{3}{9}$$

Total = $$\frac{10}{13} * \frac{4}{9} + \frac{3}{13}*\frac{3}{9}$$ = $$\frac{49}{117}$$
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Re: There are 10 women and 3 men in Room A. One person is picked  [#permalink]

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Total P ; men A * women B + women A * women a ; 3/13*3/9 + 10/13*4/9
solve
49/117 IMO B

HarishV wrote:
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?

(A) 13/21
(B) 49/117
(C) 40/117
(D) 15/52
(E) 5/18 Re: There are 10 women and 3 men in Room A. One person is picked   [#permalink] 09 Sep 2019, 08:43
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