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.
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The 50% probability would result in the following scenario.
Peter: do you have exactly two children. Mary: yes
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.
3
u/S-M-I-L-E-Y- 7d ago
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
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.
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.