Finding The Correct Probability
How tricky it was?
If you think that the probability is 1/2 after reading that the couple has equal chance of having child of either sex then you are in wrong direction.
Take a look at the table below.
There are 27 possible combinations when boy is born on Tuesday. Out of which there are only 13 possible combinations where either boy (first or second) is born on Tuesday.
Hence the probability that the person having at least 1 boy off his 2 boys born on Tuesday is 13/27.
This puzzle is usually asked a different way. Your solution is quite correct for how you phrased it, but not for how it is usually phrase. My issue is that people seldom recognize the difference.
ReplyDeleteThe puzzle originated at the 2010 "Gathering for Gardner," a math-game convention named for Martin Gardner who wrote Scientific American's "Mathematical Games" for many years. At the convention, a man walked onto the stage and announced "I have two children. One is a boy born on a Tuesday." Then he asked "What is the probability I have two boys?", and sat down.
He apparently recalled that Gardner had once asked a simpler version of the puzzle: "Mr. Smith has two children. At least one of them is a boy. What is the probability that both children are boys?" Gardner first said the answer was 1/3, but later withdrew that answer because the problem statement is ambiguous. The answer is 1/3 if, as in your version, you specifically look for a family of two that includes at least one boy.
Gardner's version can also come about if, instead of asking if at least one was a boy, you asked the random person to identify one gender among the two children. In this case, a parent of a boy and a girl would be just as likely to say "girl" as "boy". So, while there are twice as many of these families as two-boy families, only half of them would say "boy" and the answer is 1/2.
And this applies to the original version of your problem as well. We can't assume that the man was chosen for this puzzle because he had a Tuesday Boy; we can only assume that he chose the form of the statement and then picked a version of the question that applied to it. So he could have mentioned a Wednesday boy, or a Thursday Girl, in a case where he also had a Tuesday Boy.
The correct solution in either version should not merely count the combinations where "Tuesday Boy" is true, it should instead add up the probabilities of all combinations producing the statement. Of the 196 combinations, this probability is non-zero for only 27 (13 with two boys). In your version, it is 100% for all of those 27, so it is functionally equivalent to counting. My point is that being functionally equivalent is not the same thing as being correct, and this distinction affects many probability puzzles.
In the original version of the problem, the probability is 50% for 26 of combinations (12 with two boys, and 14 with a boy and a girl), and 100% for only one. Your answer is (1+12)/(1+12+14)=13/27, while the (only reasonable) answer to the original is (1+12/2)/(1+12/6+14/2)=1/2.
And the reason your version changes from 1/3 to 12/27, when you add "Born on Tuesday" to the simpler version, is because a family with two boys is almost twice as likely to answer "yes" to you question, than a family with a boy and a girl.