ANSWER:
13/27.
EXPLANATION: At first it seems the answer is 50% (or 33% if you're familiar with the simple
version of the
Two Child Problem).
Most people will be amazed that the "born on a Tuesday" detail has an effect on the probability!
The probability is calculated using Bayes' Theorem or by using a more basic approach consisting of
a table that includes all the possible combinations for the two children based on the details that were
provided. There are 196 combinations possible of gender and day of the week:
The first two characters of each combination refer to the first child and the next to characters refer
to the second child. B and G are used to indicate a boy or girl and the numbers 1 though 7 are used to
indicate the day the child was born on (Monday through Sunday). B1B1 means child 1 is a boy born on the
first day of the week (Monday) and child two is also a boy born on the first day of the week (Monday).
The combinations that include a boy born on a tuesday (B2) are highlighted, there are 27 in all and
these represent the possible combinations for the man's children given his statement. Of the 27
possibilities, 13 consist of two boys. So the probability is 13/27.
Those familiar with the classic Two Child Problem might wonder why the answer changes from 1/3 to 13/27
by adding the detail about the day of the week. By being more specific about one of his children, there
is a smaller chance that he is referring to either one of the children and a greater chance that he is
referring to one in particular. There's less "probability overlap" for the statement I guess is one way
of putting it. Another way of putting it
(described
here)
is that by adding specific information "You've effectively told people about one individual, not told
them that one of your children is a member of a category."
UPDATE: This solution (as per the puzzle's creator) is controversial. The puzzle is
discussed
here, and at the
end of the discussion, includes Gary Foshee's response to the criticism of his answer. He admits "If you
start putting in factors about how the children were chosen, from which set, then yes there is an argument
the answer could be different."
And in this
math bulletin, the
author said "the question that Foshee actually answered was:
Of all two-child families with at
least one child being a boy born on a Tuesday, what proportion of those families have two boys?
The correct answer to the question he actually posed is P = 1/2."
Ultimately, it depends on how you assume we got to the initial statement. If you assume all parents are
asked "Do you have a boy born on a Tuesday?" and then, for all those who say yes, determine the odds
they have two boys, then the 13/27 answer is correct. Whereas if an individual, who knows both of their
children, gives any set of facts about one of them, the odds that the other one is a girl/boy is still
50%. Since it is not clear how the initial statement was arrived at, in our humble opinion both answers
are correct. :-)
A man states that he has two children and that at least one of them is a boy born on a Tuesday. What
is the probability that the man has two boys? Warning: this puzzle is harder than it looks. :-)