. The probability of having a girl is identical to the probability of having a boy. In a family with three children, what is the probability that all the children are of the same gender?
a) 1/8.
b) 1/6.
c) 1/3.
d) 1/5.
e) 1/4.
OA is E
Probability

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The probability of having a girl in all 3 cases is :
=> 1/2*1/2*1/2= 1/8
The probability of NOT having a boy in all 3 cases is :
=> 1/2*1/2*1/2= 1/8
Therefore the probability of having a girl and not a boy will be;
=> 1/8 +1/8
=> 2/8
=> 1/4
Ans : 1/4
=> 1/2*1/2*1/2= 1/8
The probability of NOT having a boy in all 3 cases is :
=> 1/2*1/2*1/2= 1/8
Therefore the probability of having a girl and not a boy will be;
=> 1/8 +1/8
=> 2/8
=> 1/4
Ans : 1/4

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Well, I myself arrived at the OA in a diifernt way..
The gender of the firstborn is insignificant since we want all children to be of the same gender no matter if they are all boys or girls.
The probability for the second child to be of the same gender as the first is: ½. The same probability goes for the third child. Therefore the answer is ½ x ½ = ¼.
But, I want to see other approaches to this problem..
Thanks !
DW
The gender of the firstborn is insignificant since we want all children to be of the same gender no matter if they are all boys or girls.
The probability for the second child to be of the same gender as the first is: ½. The same probability goes for the third child. Therefore the answer is ½ x ½ = ¼.
But, I want to see other approaches to this problem..
Thanks !
DW
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Both approaches noted are great and, in a simple question like this, effective ways to solve the problem.Dream Weaver wrote:. The probability of having a girl is identical to the probability of having a boy. In a family with three children, what is the probability that all the children are of the same gender?
a) 1/8.
b) 1/6.
c) 1/3.
d) 1/5.
e) 1/4.
We could also recognize that this is a pseudo coinflip question. Any time you see 50/50 binary situations, think coin flips!
The question could have been:
if a fair coin is flipped 3 times, what's the probability of getting either exactly 3 heads or exactly 3 tails?
So, we could have applied the coin flip formula, which is:
Probability(k results out of n flips) = nCk/2^n
Since we want 3 heads OR 3 tails, we calculate the probability of each and ADD them together:
3C3/2^3 + 3C3/2^3 = 1/8 + 1/8 = 1/4
You also could have brute forced the question, much like you can with 3 coin flip questions. The possibilities are:
BBB
BBG
BGB
BGG
GBB
GBG
GGB
GGG
and 2 out of the 8 match what we desire, so 2/8 or 1/4 is the right answer.
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I used following approach. Let us say I have three letters with me and I want to arrange them.
1. All the letters are same e.g. BBB or GGG then there is only one way of arranging them.
2. Now if I have 2Bs and one G then I have three options
BBG
GBG
GBB
3. If I have 1B and two Gs so again I have three options.
BGG
GBG
GGB
Thus I can have 8 possible options
BBB
GGG
BBG
GBG
GBB
BGG
GBG
GGB
Now consider this in context of problem assume the alphabet B is boy and alphabet G is girl
We have total 8 options as listed above
Out of these 8 options two options BBB and GGG are cases where children are of same gender.
so P = no of outcomes of event/Total no. of outcomes
= 2/8 = 1/4
I am not sure whether I am clear. Please correct if something is wrong in my logic
Thanks
1. All the letters are same e.g. BBB or GGG then there is only one way of arranging them.
2. Now if I have 2Bs and one G then I have three options
BBG
GBG
GBB
3. If I have 1B and two Gs so again I have three options.
BGG
GBG
GGB
Thus I can have 8 possible options
BBB
GGG
BBG
GBG
GBB
BGG
GBG
GGB
Now consider this in context of problem assume the alphabet B is boy and alphabet G is girl
We have total 8 options as listed above
Out of these 8 options two options BBB and GGG are cases where children are of same gender.
so P = no of outcomes of event/Total no. of outcomes
= 2/8 = 1/4
I am not sure whether I am clear. Please correct if something is wrong in my logic
Thanks
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It's fine, as long as you indeed remember that there are two scenarios that need to be added together.

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The question says, the probability of all the 3 children be same gender
The children can be all 3 girls or all 3 boys.
The probability of getting a boy = 1/2
The probability of getting a girls = 1/2
Probability of getting 3 girls = 1/2 * 1/2 * 1/2 = 1/8
Probability of getting 3 boys= 1/2 * 1/2 * 1/2 = 1/8
What is asked is probability of getting of same gender i.e. either all 3 girls or all 3 boys i.e. 1/8 + 1/8 = 1/4
E is the answer
The children can be all 3 girls or all 3 boys.
The probability of getting a boy = 1/2
The probability of getting a girls = 1/2
Probability of getting 3 girls = 1/2 * 1/2 * 1/2 = 1/8
Probability of getting 3 boys= 1/2 * 1/2 * 1/2 = 1/8
What is asked is probability of getting of same gender i.e. either all 3 girls or all 3 boys i.e. 1/8 + 1/8 = 1/4
E is the answer

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Probability of having 3 girls = 1/2*1/2*1/2 = 1/8
Probability of having 3 boys = 1/2*1/2*1/2 = 1/8
Pobability of having same gender = 1/8+1/8 = 1/4
Probability of having 3 boys = 1/2*1/2*1/2 = 1/8
Pobability of having same gender = 1/8+1/8 = 1/4
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The gender of the first child is irrelevant: the first child can be either a boy or a girl.Dream Weaver wrote:. The probability of having a girl is identical to the probability of having a boy. In a family with three children, what is the probability that all the children are of the same gender?
a) 1/8.
b) 1/6.
c) 1/3.
d) 1/5.
e) 1/4.
OA is E
We care only that the second child and the third child are of the same gender as the first child.
P(2nd child is the same gender) = 1/2.
P(3rd child is the same gender) = 1/2.
Since we want both outcomes to happen, we multiply the fractions:
1/2 * 1/2 = 1/4.
The correct answer is E.
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Can you just say probability of all same gender = (1/2)^3 and this can happen in 2 ways (all boys or all girls) so 2 *1/8 = 1/4. Is this approach correct?
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This is a easy approach.....niharikamohanty@gmail.com wrote:The probability of having a girl in all 3 cases is :
=> 1/2*1/2*1/2= 1/8
The probability of NOT having a boy in all 3 cases is :
=> 1/2*1/2*1/2= 1/8
Therefore the probability of having a girl and not a boy will be;
=> 1/8 +1/8
=> 2/8
=> 1/4
Ans : 1/4
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This question is similar to a coin which is flipped 3 times. As we know the probability of a head and a tail is same ( in
This question boys and girls), we can calculate the total outcomes  2*2*2 =8
also the chances of having all boys or all girls is one each. So total favorable outcome =2
probability = 2/8=1/4
option E is the right answer
This question boys and girls), we can calculate the total outcomes  2*2*2 =8
also the chances of having all boys or all girls is one each. So total favorable outcome =2
probability = 2/8=1/4
option E is the right answer