Hey, buddies :)

Recently people had discussion in Brilliant-Lounge on a probability problem which is:

In a family of 3 children, what is the probability that at least one will be a boy?

Some of them believe that \(\frac 34\) is the correct answer while the others believe that the correct answer is \(\frac 78\).

Everyone is invited to come up with their response along with the explanation. It will be fun and help us a lot to upgrade our knowledge engine further.

Thanks!

## Comments

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TopNewestLet \(B\) represent a boy and \(G\) represent a girl.Then,

Sample Space , \(S =\{BBB,BBG,BGB,GBB,BGG,GBG,GGB,GGG\}\)

(Probability of having at least \(1\) boy) \(= 1 - \)(Probability of having only girls (no boys))=\(1-\frac{1}{8}\)(only \(1\) case out of \(8\) cases)=\(\frac{7}{8}\)

So According to me, correct answer is \(\boxed {\frac{7}{8}}\).

Note:-\(BBG,BGB,GBB\) are different cases because their relative ages (order of birth) are different in each case.

(Same for \(BGG,GBG,GGB\))

Alternate Thinking Process:-\(P(B)=P(G)=\frac{1}{2}\)

(Probability of having at least \(1\) boy) \(= 1 - \)(Probability of having only girls (no boys))\(=1-\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=\boxed{\frac{7}{8}}\) – Yash Dev Lamba · 8 months, 3 weeks ago

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@Yash Dev Lamba – Atanu Ghosh · 8 months, 3 weeks ago

Nice approachLog in to reply

– Aditya Kumar · 8 months, 3 weeks ago

Yes you are right. This is the same approach what I had.Log in to reply

The Family cares about getting Boy, not about getting a young boy or an old boy. So, how does having two younger daughters and an elder son different from having a younger son and two elder daughters ? – Vishal Yadav · 8 months, 3 weeks ago

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But there is a large assumption that boys and girls are given birth to with 0.5 probability each! That's quite huge an assumption, and it would allow any answer to be correct... – Aloysius Ng · 8 months, 3 weeks ago

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Ans should be 7/8 as if we remove the case of no boys then the remaining will be atleast 1 boy i.e 1-(1/2)^3= 7/8 ☺ – Rajat Jain · 8 months, 3 weeks ago

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Take the complementary probability, the chance that no boys are picked. For this to happen, all girls must be picked, so the probability is (1/2)^3 = 1/8. Every other case has at least one boy, so the probability that at least one boy is chosen is 1 - 1/8 = 7/8. – Alexander Koran · 8 months, 3 weeks ago

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How 3/4 will come? – Shyambhu Mukherjee · 8 months, 3 weeks ago

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– Yash Dev Lamba · 8 months, 3 weeks ago

if blindly consider BBG,BGB,GBB same and BGG,GBG,GGB also same then prob. is 3/4 which is incorrect.Log in to reply

– Shyambhu Mukherjee · 8 months, 3 weeks ago

Thanks for explaining. A wrong answer is more important than a right.Log in to reply

@Nihar Mahajan @Sharky Kesa

What do you guys think? I would be great if you participate in this discussion as you were playing a major role in the slack discussion. Thanks! – Sandeep Bhardwaj · 8 months, 3 weeks ago

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– Vishal Yadav · 8 months, 3 weeks ago

The Family cares about getting Boy, not about getting a young boy or an old boy. So, how does having two younger daughters and an elder son different from having a younger son and two elder daughters ?Log in to reply

Simple conceptual learning condtitonal probability Probability Rocks By- YDL – Yash Dev Lamba · 8 months, 3 weeks ago

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I agree with @Yash Dev Lamba

Probability can lead to amazing paradoxes. Here is a very well known probability question often misunderstood:

A family has two children. What is the probability that they are both sons, given that

a)At least one of them is a son?b)the elder child is a son? – Agnishom Chattopadhyay · 8 months, 3 weeks agoLog in to reply

(a)1/3 (b)1/2 – Harsh Shrivastava · 8 months, 3 weeks ago

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I feel like the answer is 1/5 – Lakshya Sinha · 8 months, 3 weeks ago

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– Sandeep Bhardwaj · 8 months, 3 weeks ago

Can you please give some explanation supporting your answer?Log in to reply

PS: I am weak in probability – Lakshya Sinha · 8 months, 3 weeks ago

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Don't worry. Keep practicing. You will soon be a master in combinatorics. \(\ddot \smile\) – Sandeep Bhardwaj · 8 months, 3 weeks ago

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– Pawan Pal · 8 months, 3 weeks ago

The answer is 3/4Log in to reply

– Pawan Pal · 8 months, 3 weeks ago

There are 4 possibilities - 1 boy , 2 boys , 3 boys and no boy . so at least 1 boy so the answer is 3/4 Am I correct ? Sandeep sirLog in to reply

– Sandeep Bhardwaj · 8 months, 3 weeks ago

No. The correct answer is 7/8.Log in to reply

– Pawan Pal · 8 months, 3 weeks ago

Ohkk sir I thought order won't matter .Log in to reply

– Pawan Pal · 8 months, 3 weeks ago

What I think is that BGG , GBG, GGB would be same so I count them as 1. Are we considering order of birth as well ?Log in to reply

– Kushagra Sahni · 8 months, 3 weeks ago

If we replace chilren with coins, the answer remains the same but that maybe a better way to tell you why order is necessary.Log in to reply

– Yash Dev Lamba · 8 months, 3 weeks ago

yes, we are considering although it is not mentioned in question but it is understood (I think) to consider order of birth.Log in to reply