Boy or Girl Paradox Born on Tuesday – Mind Your Decisions
There’s a math problem making the rounds on Reddit that has many people thoroughly confused.
Mary has 2 children. She tells you that one is a boy born on a Tuesday. What’s the probability the other child is a girl?
Most people think the answer is 50%. People familiar with the boy or girl paradox may guess 66.6%. But in this particular variation, the answer is about 51.8%. So people are wondering: where are the numbers coming from?
As usual, watch the video for a solution.
Boy or Girl Paradox Born on Tuesday
Or keep reading.
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Answer To Boy or Girl Paradox Born on Tuesday
(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).
If a person has a child, then according to general statistics, the next child has an equal chance of being a boy or a girl–a 50% chance like a coin flip.
But now let’s rephrase the problem. Mary has 2 children and at least one is a boy. What’s the probability the other child is a girl?
The answer is not 50%, and this is a result of conditional probability. Let’s first enumerate the equally likely events. As each child can be a boy or girl, there are 4 equally likely events:
bb
bg
gb
gg
Conditional on knowing at least one child is a boy, the event gg can be removed from the sample space so there are 3 equally likely events.
bb
bg
gb
Out of these 3 events, there are 2 in which the other child is a girl. Therefore, the probability the other child is a girl, given at least one is a boy is 2/3 = 66.6%.
Born on Tuesday variation
But now let’s rephrase the problem one more time. Mary has 2 children and at least one is a boy born on a Tuesday. What’s the probability the other child is a girl?
The child being born on a Tuesday seems like it shouldn’t matter. The boy must be born on some day of the week, after all, so why not suppose the child is born on a Tuesday.
However, this small detail affects the conditional probability calculation dramatically.
Given that at least one child is a boy, there are 3 equally likely cases to consider:
bb
bg
gb
Write a subscript to denote the day of the week a child is born, with 0 = Sun, 1 = Mon, 2 = Tues, etc. For the case of bb, suppose the second child is born on a Tuesday, so we will have 7 associated equally likely events:
b0b2
b1b2
b2b2
b3b2
b4b2
b5b2
b6b2
Now what if the first child is born on a Tuesday? We have already listed b2b2, so there are only 6 additional cases for the second child’s day of the week.
b2b0
b2b1
b2b3
b2b4
b2b5
b2b6
Then for gb we will have 7 days for g.
g0b2
g1b2
g2b2
g3b2
g4b2
g5b2
g6b2
And the same is true for bg.
b2g0
b2g1
b2g2
b2g3
b2g4
b2g5
b2g6
In total we have 7 + 6 + 7 + 7 = 27 equally likely events. There are 14 events for bg and gb, which correspond to the other child being a girl.
Therefore the probability of a girl, given at least one boy born on a Tuesday is:
14/27 = 51.8%
And that’s the surprising answer to this counter-intuitive probability problem!
References
https://www.reddit.com/r/mathmemes/comments/1nhz2i9/i_dont_get_it/
https://www.reddit.com/r/theydidthemath/comments/1ops9uu/request_how_do_they_get_to_these_numbers/
https://www.reddit.com/r/ExplainTheJoke/comments/1psit5i/shouldnt_the_odds_be_50_why_is_it_518/
Boy or girl paradox
https://en.wikipedia.org/wiki/Boy_or_girl_paradox
Limmys show
https://www.youtube.com/watch?v=-fC2oke5MFg
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