# Logic puzzle: Child gender compounded with irrelevant data

• Feb 8th 2013, 04:02 PM
AngleWyrm
Logic puzzle: Child gender compounded with irrelevant data
Quote:

Select all people who have two children, at least one of which is a boy born on Tuesday. Given one of those individuals, what is the probability that their other child is a boy?
Common sense suggests there are only two possible outcomes: TuesdayBoy+Boy, and TuesdayBoy+girl, each with a probability of 1/2.

But some hard-charging pundit for the other team has gone on to suggest otherwise by creating an outcome space like so:

 Mon-Boy Tue-Boy Wed-Boy Thu-Boy Fri-Boy Sat-Boy Sun-Boy Mon-Girl Tue-Girl Wed-Girl Thu-Girl Fri-Girl Sat-Girl Sun-Girl Mon-Boy XXXXXX Tue-Boy XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX XXXXXX Wed-Boy XXXXXX Thu-Boy XXXXXX Fri-Boy XXXXXX Sat-Boy XXXXXX Sun-Boy XXXXXX Mon-Girl XXXXXX Tue-Girl XXXXXX Wed-Girl XXXXXX Thu-Girl XXXXXX Fri-Girl XXXXXX Sat-Girl XXXXXX Sun-Girl XXXXXX

From this table of outcomes, 27/196 possible two-child families have at least one Tuesday-Boy. Furthermore, 13/196 outcomes are TuesdayBoy+Boy, while 14/196 outcomes are TuesdayBoy+girl. And I'm willing to believe that drawing a random entry from this outcome space will result in TuesdayBoy+Boy with a probability of 13/196, and TuesdayBoy+Girl with a probability of 14/196.

But I'm pretty sure the assessment that this represents the quoted question accurately is in error. And I think the mechanism of selecting the subset of qualified results is the source of that error.

Any help?
• Feb 9th 2013, 04:36 AM
tom@ballooncalculus
Re: Logic puzzle: Child gender compounded with irrelevant data
The table is as good a way as any to show that the probablity in question is 13/27.

The 'hard-charging pundit' (loving that!) just forgot the two indications ('Select all people who' and 'Given one of those') that the set of 27 is the bag out of which one of the 13 is to be drawn.
• Feb 9th 2013, 08:51 PM
AngleWyrm
Re: Logic puzzle: Child gender compounded with irrelevant data
I will agree that 13/196 possibilities in the example outcomes space are boy-boy.

But this seems like a misuse of a mathematical construct to me. Substitute any irrelevant fact into the table, and we get an equally arbitrary answer:

Quote:

I flipped a fair coin twice, and I can tell you about one of the results: One outcome was a head; also, I didn't own any cats at the time. What are the odds the other result was also a head? [P(OwnCat) declared as 1/2 my lifetime]