A woman and two kids.
A = day of the week on which kid 1 was born. B = day of the week on which kid 2 was born.
Both A and B have a uniform probability distribution on the week domain (p=1/7).
There are 7*7=49 total combinations.
The probability distribution over the combinations is still uniform (p=1/49).
Q: 'Was at least one of your kids born on Monday?' A: 'At least one of my kids was born on Monday'.
You need to figure out the new probability distribution over the combinations (p=?).
'Was at least one of your kids born on Monday?' = 'Is A=M and B=M or A=M and B not N or A not M and B=M ?' = 'Is one of these combinations? A=B B=M, A=M B=T, A=M B=W, A=M B=T, A=M B=F, A=M B=S, A=M B=S, A=T B=M, A=W B=M, A=T B=M, A=F B=M, A=S B=M, A=S B=M'.
There are only 13 allowed combinations now.
The updated probability distribution is uniform along the allowed combinations (p=1/13) and 0 everywhere else.
Q: 'On what day of the week was at least one of your kids born?' A: 'At least one of my kids was born on Monday'.
You need to figure out the new probability distribution over the combinations (p=?).
'On what day of the week was at least one of your kids born?' => Coin toss => (50% kid 1, 50% kid 2)
Both A and B have a uniform probability distribution (p=1/7), the coin has uniform probability over (head, tail).
There are 7*7*2=98 total combinations.
Before the answer is given, the probability distribution over all the 98 combinations is uniform (p=1/98).
After the answer is given, only 14 valid combinations are left.
After the answer is given, the probability distribution over the 14 valid combinations is uniform (p=1/14).
If you project the pdf in 2D, ignoring the coin toss case, you get a distribution that's not uniform anymore, as the combination MM is the sum of two valid combinations in the 3D space.
Unprompted statement: 'Hey you! At least one of my kids was born on Monday'. What's the probability distribution in this case?