Logic Puzzles

8. Two Children

I ask people at random if they have two children and also if one is a boy born on a tuesday. After a long search I finally find someone who answers yes. What is the probability that this person has two boys? Assume an equal chance of giving birth to either sex and an equal chance to giving birth on any day.

Added 1 January 2007 · Updated 5 July 2026

Hint: If I ask people at random if they have two children and if the youngest is a boy, then the probability that this person has two boys is 1/2.

If I asked people at random if they have two children and if one is a boy, then the probability that this person has two boys is 1/3.

Possibility Has at least one boy? Two boys?
BB yes yes
BG yes no
GB yes no
GG no no


My question is excluding anyone with two girls, so therefore there are only three cases left, only one of which has two boys.
Solution:

13/27. If you think the answer should be 1/2, you would be wrong. If you knew which child was a boy (say, the younger one), you would be closer to the truth. But since the boy could be either the younger or the older child, the analysis is more subtle. But what does Tuesday have to do with it?


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Comments (60)

Anonymous 15 February 2011

The question should be changed to ask if they have two children born on Tuesday and if one is a boy for the answer to make sense.

Anonymous 4 May 2011

The answer for the Two Children puzzle is 1/2, as most people guess. The correct answer for the riddle is 14/28=1/2, not 13/27.

Anonymous 23 May 2011

Both cases of the boy being born on a Tuesday are statistically independent. It would only be a valid point if it were impossible for both to be born on a Tuesday.

Anonymous 27 June 2011

The answer regarding the days of the week is incorrect. Being born on a Tuesday and the second being born on a Tuesday are independent events, which affects the probability.

Anonymous 21 July 2011

The statistical analysis assumes that once a boy had been born on Tuesday, that option is removed from the pool. In fact, Tuesday boy is still a possibility, resulting in 14/28 (1/2).

Anonymous 29 July 2011

The answer to the Two Children puzzle entails an inconsistency in interpreting the question. The phrasing suggests that the inquiry is about people having two children, but the interpretation of the probability seems flawed.

Anonymous 9 October 2011

Doesn't the solution posted for puzzle #8 reflect the gambler's fallacy? Knowing that the person has two children, one of whom is a boy born on a Tuesday, means the only relevant thing is whether the second child is a boy or a girl, which is a 50/50 probability.

Anonymous 2 November 2011

I think the answer for the 'Two Children' problem is wrong. The graph misrepresents the data given by the question, as each child needs to be represented separately. This is problematic when both children are born on the same day of the week.

Anonymous 8 December 2011

Question #8 - Two Children is ambiguous. Your solution assumed that the person had AT LEAST one boy born on a Tuesday, but one could also assume EXACTLY one boy born on a Tuesday, which would yield an answer of 12/27.

Anonymous 8 December 2011

Question #8 - Two Children is ambiguous. Your solution assumed that the person had at least one boy born on a Tuesday, which could lead to different interpretations.

Anonymous 11 January 2012

Regarding the difficult 'Two Children' puzzle, I believe the inclusion of a 7-day week is arbitrary. If we had a 10-day week, the answer would be different. The birth conditions of one child should not affect the other.

Anonymous 8 August 2012

Your two children riddle is wrong. It assumes that only one of the two could be a boy born on a Tuesday. If I ask you if one is a boy born on a Tuesday and you have two boys born on a Tuesday, then the answer is yes.

Anonymous 8 August 2012

Your riddle about two children assumes that only one could be a boy born on a Tuesday. If both are boys born on a Tuesday, then the answer is yes, one is indeed a boy born on a Tuesday. This is similar to the coin riddle where one coin is not a nickel, but the other can be.

Anonymous 3 September 2012

Two Children is wrong. It is still 50%. The person making the chart failed to recognize that the bb where both children are born on Tuesday should be counted twice, making the chances 14/28 or 50%.

Anonymous 3 September 2012

In 'Two Children', the probability should indeed be reconsidered. The scenario with both children born on Tuesday should be counted twice, affecting the overall probability.

Anonymous 3 September 2012

Concerning the difficult 'Two children' problem, I was wondering why the order of the births matters. The answer is still counterintuitive and further from one-half if order doesn't matter.

Anonymous 17 October 2012

I call BS on the Two Children one. Once one child is born, the chance of the next one is completely independent. Why does it matter if it is Tuesday? If I picked any other day of the week, would it still be 13/27?

Anonymous 17 October 2012

The Two Children puzzle's logic seems flawed. The chance of the next child being a boy or girl is independent of the first child's gender. The day of the week should not affect the probability.

Anonymous 3 November 2012

The issue is that you have the two children's charts overlapped, which would indicate they are dependent. Because they are overlapped, you are counting the column or row for the child that is already known to be a boy, except for the (Boy,Boy) (Tuesday,Tuesday) square. They are independent knowing one of them is a boy.

Anonymous 7 November 2012

Given the answer you gave, the question should read that you also ask if 'at least one' of their children is a boy born on a Tuesday. If both boys were born on Tuesday, the response to whether one was born on a Tuesday would be no because in fact there were two born on Tuesday.

Anonymous 7 November 2012

The question should clarify if 'at least one' of their children is a boy born on a Tuesday. If both boys were born on Tuesday, the response would be no because there were two born on Tuesday.

Anonymous 4 December 2012

The Two Children puzzle has an error. It's a given that one of the children is a boy, therefore there is a 100% chance that that child is a boy. The only child needed to take into consideration is the unknown because of independent assortment of odds, leading to a 50% chance that the other child is a boy.

Anonymous 4 December 2012

The solution to 'Two Children' is flawed. Given that one child is a boy, the probability of the other child being a boy is 50%, not 100%.

Anonymous 10 December 2012

Your solution is wrong. The answer is indeed 1/2. If their boy born on Tuesday is the first child, then the chance of the second child being a boy is 1/2. If their boy born on Tuesday is their second child, then the chance of the first child being a boy is also 1/2.

Anonymous 10 December 2012

The solution to the problem regarding the two children is incorrect. The probability of the second child being a boy is indeed 1/2, regardless of the day of the week.

Anonymous 5 March 2013

The error in your chart is that you counted the overlapped option only once instead of twice. There are two ways for this to happen, so it has to be counted twice. Using this, 28/56 options would be highlighted on your chart. 1/2 is the correct answer.

Anonymous 5 June 2013

I have a comment on the solution to the 'two children' problem. You state that the probability is 13/27 that the second child is a boy, but this seems doubtful to me. You show that only 13 out of 27 combinations are bb, which seems to be correct initially, but it seems that there are two options for bb.

Anonymous 5 June 2013

The probability of the second child being a boy in the 'two children' problem seems incorrect. The combinations for bb should account for the different outcomes based on the names of the boys, which may not have been fully considered.

Anonymous 18 August 2013

For Puzzle #8, if both children are boys born on Tuesday, you should only count that day once, not twice.

Anonymous 2 December 2013

The two children question has the answer wrong; it is 1 chance in two or 50%. The answer given is incorrect as it appears to ignore the fact that the second son can be born on any day of the week, therefore the day of the week is not a variable and the only factor that changes is whether it is a girl or a boy.

Anonymous 2 December 2013

The two children question has the answer wrong. It is 1 chance in two or 50%, as the second son can be born on any day of the week. The answer given ignores this fact.

Anonymous 19 January 2014

The Two Children solution contains a possibility that would mean both of them were boys born on a Tuesday, but we are told that one of them was born on a Tuesday. I got it wrong anyway.

Anonymous 9 February 2014

Puzzle 8 on the Difficult Logic Puzzles page is incorrect. The question implies that one of them is a boy born on Tuesday, not at least one. This changes the probabilities significantly.

Anonymous 1 April 2014

The probability of having two boys is still 1/2 since these are independent events. The solution provided is for the case where two boys are born on Tuesday, which is not specified in the problem.

Anonymous 7 May 2014

Number 8 on difficult in the logical puzzles section is wrong. The probability you gave for having two boys where one is born on a Tuesday is correct, but that would be in a situation where either the boys weren't born or weren't known about yet.

Anonymous 7 May 2014

The probability calculation for the two boys puzzle is incorrect; given one boy born on a Tuesday, the probability of the other child being a boy is 50%.

Anonymous 8 May 2014

I previously mentioned that the probability for puzzle #8 is wrong; the calculation of 13/27 for having two boys with one born on Tuesday is incorrect.

Anonymous 25 August 2014

The answer for 99 people saved doesn't make sense as it assumes the men in line know who is saved. The question states that any correct answer is unknown to all others. If the hats alternate colors, then everyone dies.

Anonymous 28 October 2014

#8 (Two children) is balderdash. Given a mother has two children, and given one child is a boy, the probability the second child is a boy is in fact 1/2. The graph presented is misleading because it simultaneously considers two possible cases which should be considered independently.

Anonymous 2 December 2014

I think that your "Two Children" problem is misleading. The parent confirms that one boy is born on a Tuesday, but it does not necessarily follow that the second child is not also a boy born on Tuesday.

Anonymous 13 May 2015

I believe you forgot the upper-right hand quadrant of boys born on Tuesday, so the total should be 33 and the probability should be 13/33.

Anonymous 3 October 2015

I think the probability will be 1/2 because the condition of asking if the child was born on Tuesday doesnโ€™t make a difference. The child can be born any day, and it doesn't matter if they are younger or older.

Anonymous 3 October 2015

I believe the probability of the second child being born on a Tuesday is 1/2. The condition of asking if he was born on Tuesday doesn't change the overall probability, as the child can be born any day.

Anonymous 26 October 2015

I believe the puzzle regarding two children, one of which is a boy born on Tuesday, is incorrect. I believe the answer is 1/2, not 13/27. Using the chart, the tile in which the column and row intersect, where both are boys born on Tuesday, should be counted twice.

Anonymous 29 May 2016

On the two children problem: Assuming the parents meant one or two boys born on Tuesday, we can conclude the following: The parents rolled a 14 equal-sided die twice and at least once rolled a 3.

Anonymous 25 June 2016

In your difficult logic puzzles, question 8, you say that the answer is 13/27 possibility. However, I disagree with this. If we don't consider the day of birth, there is a 1/2 chance that a newborn child will be a boy.

Anonymous 25 June 2016

I disagree with the answer of 13/27 for question 8. If we consider the possibility of having two boys without the day of birth, the chance would be 1/4. Knowing one child is a boy changes the calculation.

Anonymous 1 October 2016

The question is incorrect; the state 'BB' has a multiplicity of 2 and thus gives a different probability.

Anonymous 31 January 2017

You have ambiguity in the last question you need to eliminate: 'I ask people at random if they have two children and also if one is a boy born on a Tuesday.' What exactly does the phrase 'if one is a boy' mean?

Anonymous 14 March 2017

The probability question regarding two children and one being a boy born on a Tuesday is intriguing. However, the relevance of the day of the week should be explained further.

Anonymous 15 March 2017

The probability question about two children is interesting, but the specific details about the boy born on a Tuesday can be misleading. It would be helpful to clarify how this affects the overall probability.

Anonymous 8 September 2017

The 'two children' answer is wrong. The boy referenced in the question is either the first or second child, and there's a 50% chance the second child is male in either case.

Anonymous 8 September 2017

The answer to the 'two children' puzzle is incorrect. The boy referenced can only be either the first or second child, leading to a 50% chance for each. The graph's highlighted lines create an impossible instance where the boy is both the first and second child.

Anonymous 29 October 2017

The wording of the Two Children puzzle should be changed to clarify that it refers to children born on Tuesday, not just one being a boy born on Tuesday.

Anonymous 14 September 2018

The solution to the two children riddle is wrong. There's a possibility that both children are boys born on Tuesdays, which isn't accounted for in the solution. The real chance is 14/28 (one half) not 13/27.

Anonymous 14 September 2018

For question #8 "Two Children," the answer should be 1/2, not 13/27. The chart leaves out the "2nd" possibility of a Tuesday/Tuesday birthday scenario, which would make the chances of BB to be 14/28.

Anonymous 12 June 2019

Regarding the difficult "two children" puzzle, I was wondering if you could provide more detail as to the solution. The inclusion of a 7-day week seems arbitrary; if we had a 10-day week, the answer would be 19/39.

Anonymous 12 June 2019

The answer to the Two Children puzzle is incorrect; it should be 1/2 as most people guess. The logic used to arrive at 13/27 is not convincing.

Anonymous 15 May 2024

I think the answer for question #8 (Two Children) is incorrect. We already know that the first child is a boy. So the real question becomes: what is the probability that the second child is a boy? The probability of the second child being a boy is independent from the fact that the first child is a boy born on a Tuesday. Since the chance of giving birth to either sex is equal, the probability of the second child being a boy is 1/2 = 50%.

Anonymous 15 May 2024

I believe the answer to question #8 (Two Children) is incorrect. The probability question regarding having two boys needs clarification.

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