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Ms. Barton has four children. You are told correctly that sh

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Ms. Barton has four children. You are told correctly that sh [#permalink] New post 15 Mar 2011, 21:44
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Ms. Barton has four children. You are told correctly that she has at least two girls but you are not told which two of her four children are those girls. What is the probability that she also has two boys? (Assume that the probability of having a boy is the same as the probability of having a girl.)

(A) 1/4

(B) 3/8

(C) 5/11

(D) 1/2

(E) 6/11

[Reveal] Spoiler: OA
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Re: Ms. Barton [#permalink] New post 16 Mar 2011, 00:35
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Assume that there are 4 Ms. Barton's children in a row: 1-2-3-4. Let show all possible combinations of them being girl(G) or boy(B) giving that there at least two girls. We are seeing each child 1,2,3,4 a s a unique(so, we distinguish between them)
BBGG
BGBG
GBBG
BGGB
GBGB
GGBB

BGGG
GBGG
GGBG
GGGB
GGGG

Overall 11 variants. 6 of them satisfy the condition of 2 boys exactly (they are highlighted in bold)
Therefore the probability is 6/11
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Re: Ms. Barton [#permalink] New post 16 Mar 2011, 00:47
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Onell wrote:
Ms. Barton has four children. You are told correctly that she has at least two girls but you are not told which two of her four children are those girls. What is the probability that she also has two boys? (Assume that the probability of having a boy is the same as the probability of having a girl.)

(A) 1/4
(B) 3/8
(C) 5/11
(D) 1/2
(E) 6/11



Let me try a general approach without resorting to actually listing all possibilities (which may not be possible if we need to do similar problem for say exactly 3 out of 6 or 4 out of 8 etc.)

number of ways in which there can be two boys and two girls out of 4 children = 4!/((2!)*(2!)) = 6

Number of ways in which there will be atleast 2 girls out of 4 children = ways for 2 boys and 2 girls (6 as calculated above) + ways for 3 girls and 1 boy (=4!/3! = 4)+ways for 1 all girls (=1), so total of 11 ways.

Required probability = 6/11
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Re: Ms. Barton [#permalink] New post 16 Mar 2011, 03:20
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You need to determine the number of favourable arrangements (i.e. arrangements for which # girls = 2) out of all possible arrangements (i.e. arrangements for which # girls >= 2)

# arrangements for 2 girls and 2 boys (favourable arrangement) = 4! / (2!*2!) = 6
# arrangements for 3 girls and 1 boy = 4! / (3!*1!) = 4
# arrangements for 4 girls and 0 boys = 4! / 4! = 1

Thus, # favourable arrangements / # possible arrangements = 6 / (6 + 4+ 1) = 6/11
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Re: Ms. Barton [#permalink] New post 17 Mar 2011, 00:21
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I liked beyondgmatscore's approach a lot. Kudos !
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Re: Ms. Barton [#permalink] New post 17 Mar 2011, 20:01
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I like it too, and can be simplified even further.

There are normally 2*2*2*2=16 combinations.

We are told it is not 4B0G, and not 3B1G. It is clear that these are 1 combination, and 4 combinations (the G could be in either of 4 spots) respectively. So 16-5=11 combinations remain.

Since it is symmetric (p(g)=p(b)=0.5), there are also 5 combinations of 4G0B/3G1B, leaving 6/11.

p.s. Dont know if this is better, I never learnt all the equations with the '!' in them and when to use each, and so I always solve these intuitively.
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Re: Ms. Barton [#permalink] New post 18 Mar 2011, 09:36
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2 girls and 2 boys (favourable arrangement) = 4! / (2!*2!) = 6
3 girls and 1 boy = 4! / (3!*1!) = 4
4 girls and 0 boys = 4! / 4! = 1

favourable arrangements / possible arrangements = 6 / (6 + 4+ 1) = 6/11
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Re: Ms. Barton [#permalink] New post 22 Mar 2011, 08:24
OA IS E. Thanks guys for your help
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Re: Ms. Barton [#permalink] New post 26 Oct 2011, 12:24
Hi people,

Suddenly, something comes to my mind so awfully, but I couldn't disprove why I am wrong. Could someone help me in this.

I suddenly see it as : P(2 girls, 2 boys) = 1/2*1/2*1/2*1/2 = 1/16.(since each has a prob of 1/2).

Pls disprove me.... I don see where I am wrong, but I know surely I am.

Thanks
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Re: Ms. Barton [#permalink] New post 26 Oct 2011, 19:33
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Expert's post
raghupara wrote:
Hi people,

Suddenly, something comes to my mind so awfully, but I couldn't disprove why I am wrong. Could someone help me in this.

I suddenly see it as : P(2 girls, 2 boys) = 1/2*1/2*1/2*1/2 = 1/16.(since each has a prob of 1/2).

Pls disprove me.... I don see where I am wrong, but I know surely I am.

Thanks


The following is the complete explanation for this question:

You need to find Probability of 2 boys and 2 girls given there are at least 2 girls. This is a conditional probability. You have already been given that there are at least 2 girls. You need to find the probability of 2 boys and 2 girls given this condition.
P(A given B) = P(A)/P(B)

P(2 boys, 2 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/(2!*2!) = 3/8
You multiply by 4!/(2!*2!) because out of the four children, any 2 could be boys and the other two would be girls. So you have to account for all arrangements: BGBG, BBGG, BGGB etc

P(atleast 2 girls) = P(2 boys, 2 girls) + P(1 boy, 3 girls) + P(4 girls) = 1 - P(4 boys) - P(3 boys, 1 girl)
You can solve P(atleast 2 girls) in any way you like.
P(1 boy, 3 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/3! = 1/4
P(4 girls) = (1/2)*(1/2)*(1/2)*(1/2) = 1/16
P(atleast 2 girls) = 3/8 + 1/4 + 1/16 = 11/16

P(2 boys, 2 girls given atleast 2 girls) = P(2 boys, 2 girls) / P(atleast 2 girls) = (3/8)/(11/16) = 6/11

In the above given solutions, all the (1/2)s have been ignored because the probability of a boy is 1/2 and of a girl is 1/2 too. Hence in every case you will get (1/2)*(1/2)*(1/2)*(1/2) which can be ignored. We just need to focus on arrangements in each case. This is similar to the famous coin flipping questions (the probability of 3 Heads and a tail etc) (check out Binomial Probability in Veritas Combinatorics book for an explanation)
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Re: Ms. Barton [#permalink] New post 26 Oct 2011, 20:58
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4 girls, 0 boys - 4C4=1
3 girls, 1 boy-4C3=4
2 girls, 2 boys -4C2=6
total- 6+4+1=11

so ,the probability of "2 boys" is 6/11
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Re: Ms. Barton [#permalink] New post 26 Oct 2011, 23:54
VeritasPrepKarishma wrote:
raghupara wrote:
Hi people,

Suddenly, something comes to my mind so awfully, but I couldn't disprove why I am wrong. Could someone help me in this.

I suddenly see it as : P(2 girls, 2 boys) = 1/2*1/2*1/2*1/2 = 1/16.(since each has a prob of 1/2).

Pls disprove me.... I don see where I am wrong, but I know surely I am.

Thanks


The following is the complete explanation for this question:

You need to find Probability of 2 boys and 2 girls given there are at least 2 girls. This is a conditional probability. You have already been given that there are at least 2 girls. You need to find the probability of 2 boys and 2 girls given this condition.
P(A given B) = P(A)/P(B)

P(2 boys, 2 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/(2!*2!) = 3/8
You multiply by 4!/(2!*2!) because out of the four children, any 2 could be boys and the other two would be girls. So you have to account for all arrangements: BGBG, BBGG, BGGB etc

P(atleast 2 girls) = P(2 boys, 2 girls) + P(1 boy, 3 girls) + P(4 girls) = 1 - P(4 boys) - P(3 boys, 1 girl)
You can solve P(atleast 2 girls) in any way you like.
P(1 boy, 3 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/3! = 1/4
P(4 girls) = (1/2)*(1/2)*(1/2)*(1/2) = 1/16
P(atleast 2 girls) = 3/8 + 1/4 + 1/16 = 11/16

P(2 boys, 2 girls given atleast 2 girls) = P(2 boys, 2 girls) / P(atleast 2 girls) = (3/8)/(11/16) = 6/11

In the above given solutions, all the (1/2)s have been ignored because the probability of a boy is 1/2 and of a girl is 1/2 too. Hence in every case you will get (1/2)*(1/2)*(1/2)*(1/2) which can be ignored. We just need to focus on arrangements in each case. This is similar to the famous coin flipping questions (the probability of 3 Heads and a tail etc) (check out Binomial Probability in Veritas Combinatorics book for an explanation)



Thanks VeritasPrepKarisma, for simplifying it so much.
Kudos for the details!
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Re: Ms. Barton [#permalink] New post 13 Nov 2011, 13:05
I got E. This question is a variation of the boy girl paradox
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Re: Ms. Barton has four children. You are told correctly that sh [#permalink] New post 23 Oct 2013, 16:30
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Re: Ms. Barton [#permalink] New post 16 Nov 2013, 01:52
VeritasPrepKarishma wrote:
raghupara wrote:
Hi people,

Suddenly, something comes to my mind so awfully, but I couldn't disprove why I am wrong. Could someone help me in this.

I suddenly see it as : P(2 girls, 2 boys) = 1/2*1/2*1/2*1/2 = 1/16.(since each has a prob of 1/2).

Pls disprove me.... I don see where I am wrong, but I know surely I am.

Thanks


The following is the complete explanation for this question:

You need to find Probability of 2 boys and 2 girls given there are at least 2 girls. This is a conditional probability. You have already been given that there are at least 2 girls. You need to find the probability of 2 boys and 2 girls given this condition.
P(A given B) = P(A)/P(B)

P(2 boys, 2 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/(2!*2!) = 3/8
You multiply by 4!/(2!*2!) because out of the four children, any 2 could be boys and the other two would be girls. So you have to account for all arrangements: BGBG, BBGG, BGGB etc

P(atleast 2 girls) = P(2 boys, 2 girls) + P(1 boy, 3 girls) + P(4 girls) = 1 - P(4 boys) - P(3 boys, 1 girl)
You can solve P(atleast 2 girls) in any way you like.
P(1 boy, 3 girls) = (1/2)*(1/2)*(1/2)*(1/2) * 4!/3! = 1/4
P(4 girls) = (1/2)*(1/2)*(1/2)*(1/2) = 1/16
P(atleast 2 girls) = 3/8 + 1/4 + 1/16 = 11/16

P(2 boys, 2 girls given atleast 2 girls) = P(2 boys, 2 girls) / P(atleast 2 girls) = (3/8)/(11/16) = 6/11

In the above given solutions, all the (1/2)s have been ignored because the probability of a boy is 1/2 and of a girl is 1/2 too. Hence in every case you will get (1/2)*(1/2)*(1/2)*(1/2) which can be ignored. We just need to focus on arrangements in each case. This is similar to the famous coin flipping questions (the probability of 3 Heads and a tail etc) (check out Binomial Probability in Veritas Combinatorics book for an explanation)


Hi Karishma
I have two questions.
1. when calculating 2 boys and 2 girls, given that there are two girls, why do we multiply 4!/2!2! with (1/2)^4?
We already know that two are girls, so shouldn't we multiply only by (1/2)^2?
2. When considering the overall, why do we divide (1) by p(at least two girls)?
Why don't we divide it by all available options?

Thanks
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Re: Ms. Barton has four children. You are told correctly that sh [#permalink] New post 04 Dec 2013, 19:00
What am I missing here?

The questions clearly states that 2 of the 4 kids are girls. It also clearly asks "what is the probability that the other 2 are boys?"

This gives us a P of 1/2 that a child will be a boy and 1/2 that the child will be a girl. Armed with the fact that having two girls will have no significance on the sex of the other two children, we can treat this as mutually exclusive scenario.

So there is P:1/2 that one of the remaining children is a boy, and P:1/2 that the other child is also a boy. Therefore, the probability would become 1/4.

Am I reading this question wrong?
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Re: Ms. Barton has four children. You are told correctly that sh [#permalink] New post 04 Dec 2013, 20:10
Expert's post
forevertfc wrote:
What am I missing here?

The questions clearly states that 2 of the 4 kids are girls. It also clearly asks "what is the probability that the other 2 are boys?"

This gives us a P of 1/2 that a child will be a boy and 1/2 that the child will be a girl. Armed with the fact that having two girls will have no significance on the sex of the other two children, we can treat this as mutually exclusive scenario.

So there is P:1/2 that one of the remaining children is a boy, and P:1/2 that the other child is also a boy. Therefore, the probability would become 1/4.

Am I reading this question wrong?


We are concerned about the 4 selections together, not the individual selection. What this means is that if your logic is correct, the probability of having 4 girls would be the same as the probability of having 2 girls and 2 boys. But is that correct? No it is not.

In how many ways can you have 4 girls?
(1/2)*(1/2)*(1/2)*(1/2) = 1/16 (First child girl,next girl, next girl, next girl)

In how many ways can you have 2 girls and 2 boys?
(1/2)*(1/2)*(1/2)*(1/2) = 1/16 (First child girl,next girl, next boy, next boy)
(1/2)*(1/2)*(1/2)*(1/2) = 1/16 (First child girl,next boy, next girl, next boy)
(1/2)*(1/2)*(1/2)*(1/2) = 1/16 (First child girl,next boy, next boy, next girl)
and so on...

There is a sequence to obtaining kids and that needs to be factored in. Many ways will lead to 2 girls and 2 boys hence the probability of obtaining 2 girls and 2 boys will be higher than the probability of obtaining 4 girls. The question clearly tells us that we don't know which 2 are girls so we cannot say that this the case where first child and second child are girls and others are boys. Hence you need to follow the method discussed above: ms-barton-has-four-children-you-are-told-correctly-that-sh-110962.html?sid=9797da5586e03fc5fd356ef7e6808458#p991399

It is similar to the coin flipping case. Every coin flip is independent of the other but the probability of obtaining 4 heads in 4 flips is less than the probability of obtaining 2 heads and 2 tails.
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Re: Ms. Barton has four children. You are told correctly that sh   [#permalink] 04 Dec 2013, 20:10
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