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# In a room filled with 7 people, 4 people have exactly 1 sibl

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Manager
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In a room filled with 7 people, 4 people have exactly 1 sibl [#permalink]

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09 Jul 2009, 19:39
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In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?

A. 5/21
B. 3/7
C. 4/7
D. 5/7
E. 16/21

OPEN DISCUSSION OF THIS QUESTION IS HERE: in-a-room-filled-with-7-people-4-people-have-exactly-87550.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Sep 2013, 01:41, edited 1 time in total.
Renamed the topic, edited the question and added the OA.

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Re: Probability - Siblings in the room [#permalink]

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10 Jul 2009, 03:47
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I3igDmsu wrote:
In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?
5/21
3/7
4/7
5/7
16/21

[Reveal] Spoiler:
OA=E

I divide 7 people into 2 groups: Group 1 including 4 people have exactly 1 sibling and Group 2 including 3 people have exactly 2 siblings
First person:
If he is from G1 (probability: 4/7), the probability that the second one is not his sibling: 5/6 so 4/7*5/6=10/21
If he is from G2 (probability: 3/7), the probability that the second one is not his sibling: 4/6 so 3/7*4/6=6/21
10/21+6/21=16/21

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Re: Probability - Siblings in the room [#permalink]

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10 Jul 2009, 09:26
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I have a basic question:
If A is a sibling of B and B is a sibling of C, do we consider A as a sibling of C? Then there should be more than 7 ppl in the room.

If we don't assume the above, then I get 4/7.
We have two mutually exclusive groups. A grp of 4 where each has one sibling, forms a square with vertices as ppl and edges as relationship.

A grp of 3 where each has two siblings, forms a triangle with vertices as ppl and edges as relationship.

Basically, we have to get one guy from G1 and other from G2.
So , I get, 4C1 * 3C1 / 7C2 = 4/7

Any comments? explanations?
DavidArchuleta wrote:
I3igDmsu wrote:
In a room filled with 7 people, 4 people have exactly 1 sibling in the room and 3 people have exactly 2 siblings in the room. If two individuals are selected from the room at random, what is the probability that those two individuals are NOT siblings?
5/21
3/7
4/7
5/7
16/21

[Reveal] Spoiler:
OA=E

I divide 7 people into 2 groups: Group 1 including 4 people have exactly 1 sibling and Group 2 including 3 people have exactly 2 siblings
First person:
If he is from G1 (probability: 4/7), the probability that the second one is not his sibling: 5/6 so 4/7*5/6=10/21
If he is from G2 (probability: 3/7), the probability that the second one is not his sibling: 4/6 so 3/7*4/6=6/21
10/21+6/21=16/21

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Re: Probability - Siblings in the room [#permalink]

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10 Jul 2009, 09:48
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If 4 people have exactly 1 sibling, there are 2 pairs of siblings in the 4 people

Say A B C and D, each have exactly 1 sibling. That implies, A has 1 sibling (say B), so B's sibling is A
Similarly, C and D are siblings

Now if 3 people (say E F and G) each have 2 siblings, then E must have F and G both as siblings
F must have E and G both as siblings, and same for G too.

So we have 2 pairs of 2 siblings and 1 triplet of 3 siblings
A-B
C-D
E-F-G

ways of selecting 2 out of 7 is = 21 ways
ways of NOT selecting a sibling pair = Total ways - ways of always getting a pair

ways of getting a sibling pair :
A-B = 1way
C-D = 1 way
2 out of 3 (E - F - G) = 3 ways

so 5 ways we always get a sibling pair, remaining 21-5 ways we dont get a sibling pair

ans is 16/21

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Re: Probability - Siblings in the room [#permalink]

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10 Jul 2009, 10:24
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awesome...+1

arghhh..i was only thinking about one way relationship..A-B = B-A and hence we have two pairs..true

rashminet84 wrote:
If 4 people have exactly 1 sibling, there are 2 pairs of siblings in the 4 people

Say A B C and D, each have exactly 1 sibling. That implies, A has 1 sibling (say B), so B's sibling is A
Similarly, C and D are siblings

Now if 3 people (say E F and G) each have 2 siblings, then E must have F and G both as siblings
F must have E and G both as siblings, and same for G too.

So we have 2 pairs of 2 siblings and 1 triplet of 3 siblings
A-B
C-D
E-F-G

ways of selecting 2 out of 7 is = 21 ways
ways of NOT selecting a sibling pair = Total ways - ways of always getting a pair

ways of getting a sibling pair :
A-B = 1way
C-D = 1 way
2 out of 3 (E - F - G) = 3 ways

so 5 ways we always get a sibling pair, remaining 21-5 ways we dont get a sibling pair

ans is 16/21

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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 05:21
Im missing something.

groups are
a-b
c-d
e-f-g

total options 21

if i choose A - i can choose c-d-e-f-g (5 options)
if i choose B - i can choose c-d-e-f-g (5 options)
if i choose C - i can choose a-b-e-f-g (5 options)
if i choose D - i can choose a-b-e-f-g (5 options)

and more. if i choose one from the triple group.so what am i missing?
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 05:42
144144 wrote:
Im missing something.

groups are
a-b
c-d
e-f-g

total options 21

if i choose A - i can choose c-d-e-f-g (5 options)
if i choose B - i can choose c-d-e-f-g (5 options)
if i choose C - i can choose a-b-e-f-g (5 options)
if i choose D - i can choose a-b-e-f-g (5 options)

and more. if i choose one from the triple group.so what am i missing?

Nothing. Keep going the way you started. You will end up with 16 favorable cases.

P=16/21

OR

Method1(Combination):

Group1: a,b
Group2: c,d
Group3: e,f,g

Choose 1 from Group1 AND Choose 1 from Group2
OR
Choose 1 from Group2 AND Choose 1 from Group3
OR
Choose 1 from Group1 AND Choose 1 from Group3

We know AND=* and OR=+

Favorable cases:
C(2,1)*C(2,1)+C(2,1)*C(3,1)+C(2,1)*C(3,1)=2*2+2*3+2*3=4+6+6=16
Total
C(7,2)=21
P=16/21

Method2(Probability):
2/7*5/6+2/7*5/6+3/7*4/6=(1/42)*(10+10+12)=32/42=16/21

Ans: "E"
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 06:12
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can we ever expect so tough questions in GMAT ?

wondering if I can ever get it

Anyways thanks for the all the replies...helps in devloping line of thinking

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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 09:05
Other way - Probability that 2 individuals chosen are siblings-
from 3 groups , A- B, C-D, E-F-G
2 * 1 / 7 * 6 + 2 * 1 / 7 * 6 + 3 * 2/ 7 * 6 = 10 / 7 * 6
= 5/21
Are not siblings = 1-5/21
= 16/21

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Manager
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 09:42
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agdimple333 wrote:
Other way - Probability that 2 individuals chosen are siblings-
from 3 groups , A- B, C-D, E-F-G
2 * 1 / 7 * 6 + 2 * 1 / 7 * 6 + 3 * 2/ 7 * 6 = 10 / 7 * 6
= 5/21
Are not siblings = 1-5/21
= 16/21

Lets put the equaltion step wise

Probability that 2 individuals chosen are siblings-
from 3 groups , A- B, C-D, E-F-G
(2 / 7) * (1/6) + (2 / 7) * (1/6) + (3/7) * (2/6) = 5/21

Are not siblings = 1-5/21
= 16/21
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 10:07
fluke wrote:
144144 wrote:
Im missing something.

groups are
a-b
c-d
e-f-g

total options 21

if i choose A - i can choose c-d-e-f-g (5 options)
if i choose B - i can choose c-d-e-f-g (5 options)
if i choose C - i can choose a-b-e-f-g (5 options)
if i choose D - i can choose a-b-e-f-g (5 options)

and more. if i choose one from the triple group.so what am i missing?

Nothing. Keep going the way you started. You will end up with 16 favorable cases.

P=16/21

OR

Method1(Combination):

Group1: a,b
Group2: c,d
Group3: e,f,g

Choose 1 from Group1 AND Choose 1 from Group2
OR
Choose 1 from Group2 AND Choose 1 from Group3
OR
Choose 1 from Group1 AND Choose 1 from Group3

We know AND=* and OR=+

Favorable cases:
C(2,1)*C(2,1)+C(2,1)*C(3,1)+C(2,1)*C(3,1)=2*2+2*3+2*3=4+6+6=16
Total
C(7,2)=21
P=16/21

Method2(Probability):
2/7*5/6+2/7*5/6+3/7*4/6=(1/42)*(10+10+12)=32/42=16/21

Ans: "E"

Im already at 20 possibilities without even the 3rd group. 5+5+5+5 (=20) out of 21 possible options. so its not working. I am missing something here.
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 10:21
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144144 wrote:
Im already at 20 possibilities without even the 3rd group. 5+5+5+5 (=20) out of 21 possible options. so its not working. I am missing something here.

For your sake:
a,b
c,d
e,f,g

----
a,c
a,d
a,e
a,f
a,g
----
b,c
b,d
b,e
b,f
b,g
----
c,e
c,f
c,g
----
d,e
d,f
d,g
----
Total=16

The 5 combinations that should be excluded:
a,b
c,d
e,f
e,g
f,g
----------------
Total=5
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 10:30
OMG. im so stupid. I guess i need to go to sleep for today. I was counting them twice.

thanks. +1
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Re: Probability - Siblings in the room [#permalink]

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16 May 2011, 18:25
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Alchemist1320 wrote:
can we ever expect so tough questions in GMAT ?

wondering if I can ever get it

Anyways thanks for the all the replies...helps in devloping line of thinking

The question is not very difficult. Try to take one line at a time to reason it out. I will tell you what I mean. I read the first line:

"In a room filled with 7 people, 4 people have exactly 1 sibling in the room and"

I stop here. 4 people have exactly 1 sibling. So I say to myself, "Ok. A is there and he has a sibling B. Wait, this means that automatically B has the sibling A. They cannot have any other sibling since both should have only one sibling each. Then out of 4 people, 2 are already accounted for. Then in the same way, there must be C there and his sibling D". The important thing to realize is that 'sibling' relation is symmetric. If A is sibling of B, B is also the sibling of A.

Next line of the question:
"3 people have exactly 2 siblings in the room."
Now I think, "There is E there and he has two siblings F and G. So automatically F and G both have 2 siblings each too."

This gives me following siblings:
A - B
C - D
E - F - G

Now I need to pick 2 people such that they are not siblings. I can do it in 2 ways. I can either find the number of ways of picking siblings or number of ways of picking 'non siblings'. Number of ways of picking siblings is certainly easier - I can pick A-B or C-D or 2 of E-F-G in 3 ways (E-F or F-G or E-G) so there are a total of 5 ways of picking siblings.
Total ways of picking 2 people is 7C2 = 21 ways
Out of these 21 ways, 5 ways are there to pick 2 siblings so other 16 ways must be of picking 'non siblings'.

Required probability = 16/21
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Re: Probability - Siblings in the room [#permalink]

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21 Jun 2011, 22:38
VeritasPrepKarishma wrote:
Alchemist1320 wrote:
can we ever expect so tough questions in GMAT ?

wondering if I can ever get it

Anyways thanks for the all the replies...helps in devloping line of thinking

The question is not very difficult. Try to take one line at a time to reason it out. I will tell you what I mean. I read the first line:

"In a room filled with 7 people, 4 people have exactly 1 sibling in the room and"

I stop here. 4 people have exactly 1 sibling. So I say to myself, "Ok. A is there and he has a sibling B. Wait, this means that automatically B has the sibling A. They cannot have any other sibling since both should have only one sibling each. Then out of 4 people, 2 are already accounted for. Then in the same way, there must be C there and his sibling D". The important thing to realize is that 'sibling' relation is symmetric. If A is sibling of B, B is also the sibling of A.

Next line of the question:
"3 people have exactly 2 siblings in the room."
Now I think, "There is E there and he has two siblings F and G. So automatically F and G both have 2 siblings each too."

This gives me following siblings:
A - B
C - D
E - F - G

Now I need to pick 2 people such that they are not siblings. I can do it in 2 ways. I can either find the number of ways of picking siblings or number of ways of picking 'non siblings'. Number of ways of picking siblings is certainly easier - I can pick A-B or C-D or 2 of E-F-G in 3 ways (E-F or F-G or E-G) so there are a total of 5 ways of picking siblings.
Total ways of picking 2 people is 7C2 = 21 ways
Out of these 21 ways, 5 ways are there to pick 2 siblings so other 16 ways must be of picking 'non siblings'.

Required probability = 16/21

Can someone please explain this ?

I can pick A-B or C-D or 2 of E-F-G in 3 ways (E-F or F-G or E-G) so there are a total of 5 ways of picking siblings.
Total ways of picking 2 people is 7C2 = 21 ways
Out of these 21 ways, 5 ways are there to pick 2 siblings so other 16 ways must be of picking 'non siblings'.

Required probability = 16/21[/quote]

And also, How do we get 7C2?? Are there 7 possible cases?

A-B
C-D
E-F
F-G
E-G

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Re: Probability - Siblings in the room [#permalink]

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21 Jun 2011, 22:49
siddhans wrote:
VeritasPrepKarishma wrote:
Alchemist1320 wrote:
can we ever expect so tough questions in GMAT ?

wondering if I can ever get it

Anyways thanks for the all the replies...helps in devloping line of thinking

The question is not very difficult. Try to take one line at a time to reason it out. I will tell you what I mean. I read the first line:

"In a room filled with 7 people, 4 people have exactly 1 sibling in the room and"

I stop here. 4 people have exactly 1 sibling. So I say to myself, "Ok. A is there and he has a sibling B. Wait, this means that automatically B has the sibling A. They cannot have any other sibling since both should have only one sibling each. Then out of 4 people, 2 are already accounted for. Then in the same way, there must be C there and his sibling D". The important thing to realize is that 'sibling' relation is symmetric. If A is sibling of B, B is also the sibling of A.

Next line of the question:
"3 people have exactly 2 siblings in the room."
Now I think, "There is E there and he has two siblings F and G. So automatically F and G both have 2 siblings each too."

This gives me following siblings:
A - B
C - D
E - F - G

Now I need to pick 2 people such that they are not siblings. I can do it in 2 ways. I can either find the number of ways of picking siblings or number of ways of picking 'non siblings'. Number of ways of picking siblings is certainly easier - I can pick A-B or C-D or 2 of E-F-G in 3 ways (E-F or F-G or E-G) so there are a total of 5 ways of picking siblings.
Total ways of picking 2 people is 7C2 = 21 ways
Out of these 21 ways, 5 ways are there to pick 2 siblings so other 16 ways must be of picking 'non siblings'.

Required probability = 16/21

Can someone please explain this ?

I can pick A-B or C-D or 2 of E-F-G in 3 ways (E-F or F-G or E-G) so there are a total of 5 ways of picking siblings.
Total ways of picking 2 people is 7C2 = 21 ways
Out of these 21 ways, 5 ways are there to pick 2 siblings so other 16 ways must be of picking 'non siblings'.

Required probability = 16/21

And also, How do we get 7C2?? Are there 7 possible cases?

A-B
C-D
E-F
F-G
E-G[/quote]

7C2 because there are 7 people in the room and you need to select 2 individuals..

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Re: Probability - Siblings in the room [#permalink]

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21 Jun 2011, 22:53
sudhir18n wrote:
7C2 because there are 7 people in the room and you need to select 2 individuals..

Ah,, actually i re-read it after i posted and understood... Thanks

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Re: Probability - Siblings in the room [#permalink]

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21 Jun 2011, 23:03
siddhans wrote:
agdimple333 wrote:
Other way - Probability that 2 individuals chosen are siblings-
from 3 groups , A- B, C-D, E-F-G
2 * 1 / 7 * 6 + 2 * 1 / 7 * 6 + 3 * 2/ 7 * 6 = 10 / 7 * 6
= 5/21
Are not siblings = 1-5/21
= 16/21

Hi sudhir,

Can you explain this method too?

okai .. here it goes,
the siblings are A-B, C-D and E-F-G
selecting 2 siblings from a,b : first can be done in 2/7 ways , second in 1/6 = 2/7*1/6 = 2/42
selecting 2 sibling from c,d : first in 2/7 ways , second in 1/6 = 2/42
selecting 2 siblings from e f g : first in 3/42 ways second in 2/42 ways = 6/42 ways

total ways to select siblings = 2/42+2/42+6/42 = 10/42 = 5/21

ways of not selecting sibling = 1-5/21 = 16/21

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Re: Probability - Siblings in the room [#permalink]

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21 Jun 2011, 23:09
sudhir18n wrote:
siddhans wrote:
agdimple333 wrote:
Other way - Probability that 2 individuals chosen are siblings-
from 3 groups , A- B, C-D, E-F-G
2 * 1 / 7 * 6 + 2 * 1 / 7 * 6 + 3 * 2/ 7 * 6 = 10 / 7 * 6
= 5/21
Are not siblings = 1-5/21
= 16/21

Hi sudhir,

Can you explain this method too?

okai .. here it goes,
the siblings are A-B, C-D and E-F-G
selecting 2 siblings from a,b : first can be done in 2/7 ways , second in 1/6 = 2/7*1/6 = 2/42
selecting 2 sibling from c,d : first in 2/7 ways , second in 1/6 = 2/42
selecting 2 siblings from e f g : first in 3/42 ways second in 2/42 ways = 6/42 ways

total ways to select siblings = 2/42+2/42+6/42 = 10/42 = 5/21

ways of not selecting sibling = 1-5/21 = 16/21

You mean selecting 2 siblings from 7 people can be done in 2/7 ways and after selecting one sibling there are 6 people left so selecting 1 from them will be 1/6 ??? I am still not clear about 2/7 and 1/6?? Sorry

Also, you take combinations when order doesnt matter ? Correct? so in 7c2 order didnt matter A-C or C-A is same?

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Re: Probability - Siblings in the room [#permalink]

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22 Jun 2011, 00:16
Dont get confused by multiple approaches mentioned in this thread. We can use both permutations and combinations.
7C2- was using combination approach.
but in the above example we have used permutations..
totsal is 7P2 = 42.

You mean selecting 2 siblings from 7 people can be done in 2/7 ways and after selecting one sibling there are 6 people left so selecting 1 from them will be 1/6 ??? I am still not clear about 2/7 and 1/6??

yes ur right !! selecting one out of six but that one should be from the already slected pair of siblings. So once we select a in 2/7 ways , we can only chose b (no one else) in 1/6 ways

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Re: Probability - Siblings in the room   [#permalink] 22 Jun 2011, 00:16

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