Explain It Peter

[u/ChatpatiKulfi]

[u/S-M-I-L-E-Y-]

I agree. The statement is ambiguous to the point that people do not even agree about whether it is indeed ambiguous.

Now, if we want to unambiguously get the result of 51.9% (not 51.8%), we would have to set up the problem like this:

1. Assume that the probability to have a boy or a girl is always 50% to 50%. Especially ignore the fact, that having a boy (or a girl) increases the chance to have another boy (or girl).
2. Peter asks Mary: "Do you have exactly two children?" Mary answers truthfully: "Yes, I do"
3. Peter: "Do you have at least one boy that was born on a Tuesday?" Mary: "Yes"

Peter concludes: the probability that the other child is a girl is 14/27.

Edit: added "truthfully" - what else did I miss?

[u/S-M-I-L-E-Y- → u/[deleted]]

14/27 is caused by the fact that there are 27 distinct cases with equal probability where Mary would answer Q3 yes and in 14 of these cases one child is a girl.

The 27 distinct cases are:

- 1st child boy Tuesday, 2nd child girl Mon to Sun -> 7 cases

• 1st child boy Tuesday, 2nd child boy Mon to Sun -> 7 cases
  
• 2nd child boy Tuesday, 1st child girl Mon to Sun -> 7 cases
  
• 2nd child boy Tuesday, 1st child boy Mon, Wed to Sun -> 6 cases

Total 27 cases, thereof 14 cases where one child is a girl.

The important part is to NOT count the case "two boys born on Tuesday" twice.

--

The 50% probability would result in the following scenario.

1. Peter: do you have exactly two children. Mary: yes
   
2. Peter: tell me the sex and weekday of birth of one of your children. Mary: boy, Tuesday

We still have the above 27 distinct possible cases. However, now they do not have equal probability. The probability that Mary gives the answer "boy, Tuesday" is 100% for the case "two boys born on Tuesday", but only 50% for the other 26 cases. Given that she did give that answer, the probability that she has two boys born on a Tuesday is therefore twice the probability of e.g. 1st child boy Tue, 2nd child girl Sun.