Making Money from Tuesday's Child

Here's a way to make money from the Tuesday's Child problem, assuming you can convince people that John Derbyshire is right and the probability in question is 13/27 (or 0.48, nearly even odds).

First, we need to convince seven of your neighbors of Derb's analysis, to wit, that the probability in question is 13/27. Then we need to find a large number of fathers with two children, at least one of whom is a boy. It doesn't matter on what days they were born. Have them bring the birth certificates. I hope you will agree with me that the probability that any father in this group has two boys is 1/3 (or 0.333, not 0.48).

Then we segregate the fathers into seven groups according to what day of the week the boy was born on. If the father has two boys born on different days of the week, have him flip a coin and join son #1's group if heads, son #2's group if tails.

Have the group with a boy born on Tuesday line up, and then repeatedly knock on neighbor #1's door and say:

"I have two children. At least one is a boy. He was born on Tuesday. Two dollars will get you three if I have two boys."

Hopefully neighbor #1 will take the bet, having been convinced that he's getting good odds. He's getting a 3:2 payout on a bet that he thinks is approximately 1:1.

Have the group with a boy born on Wednesday line up, and then repeatedly knock on neighbor #2's door and say:

"I have two children. At least one is a boy. He was born on Wednesday. Two dollars will get you three if I have two boys."

Do this with the groups for the remaining days of the week and remaining neighbors. As far as we are concerned, we are paying out at 3:2 a bet that is actually 2:1! (A fair payout would be four dollars for every two dollars bet. We are paying out as though the odds were 2/5 or 0.40. Our advantage is 7%, better than the house advantage in a typical casino game. ) 

As far as any single neighbor is concerned, he's simply seen a repetition of the Tuesday Child problem. Everybody who knocks on his door says the same day of the week. Once you've got all their money, they might get suspicious, but you've got the birth certificates to back it up... and the (bogus) mathematical analysis. Some people are just lucky, you will tell them.

The Tuesday's Child Game

In this post I discussed the so-called Tuesday's Child problem in probability theory. The theory (with which I disagree) is that the probability of that the speaker has two boys is 13/27. I think the probability is actually 1/3. I've made a crude Java applet, the Tuesday's Child Game, to demonstrate the point.

Since the theory claims the odds are approximately even that the speaker has two boys, if I give you anything better than even odds, you are probabilistically ahead of the game. The applet pays off at 3:2 so,
according to the theory, you should win a lot of money.

If you are still interested in this problem, and can get $100 before going broke, let me know.

Tuesday's Child

Here is a post at the corner concerning a probability question. It leads to a number of other posts and to Derbyshire's analysis here. If you follow the chain of posts back, it leads to other sites and considerable debate over the interpretation of this problem.

The analysis is tricky only if the problem is interpreted in something other than its straightforward, plain meaning. The statement of the problem is:

"I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?"

Now the problems all come from that "born on a Tuesday" clause in the middle sentence. Take that out, and everyone agrees on the answer:

"I have two children. One is a boy. What is the probability I have two boys?"

This is the classic coin-flip enumeration problem. Having children is like flipping a coin, with heads = boys and tails = girls. The possible outcomes for two consecutive coin flips are:

Heads - Heads
Heads - Tails
Tails - Heads
Tails - Tails

or, in the boy girl terms:

Boy - Boy
Boy - Girl
Girl - Boy
Girl - Girl

Since we know that at least one of the children is a boy, the last case is ruled out and the probability that the speaker has two boys is one in three.

Returning to the original problem, the analysts all seem to think the phrase "born on a Tuesday" is very significant, but they can't agree on its significance. I don't think it adds anything to the problem at all. In the straightforward, obvious interpretation, it is only a statement after the fact of birth concerning the day of birth. It's like saying the boy was 8 lbs at birth, or was born with blue eyes. It doesn't say anything about the prior possibilities of weight or birthdays; it is only a statement about what in fact occurred. It doesn't say that one or both boys couldn't have been born on a Wednesday. If that had happened, the consequence would be that the problem would say:

"I have two children. One is a boy born on a Wednesday. What is the probability I have two boys?"

The answer to this question is the same as the answer to the Tuesday question and to the simpler question that does not refer to a day at all: 1/3.

Derbyshire calculates the probability as 13/27. He can only get there by interpreting the "Tuesday clause" as affecting the prior probabilities of birth. In other words, the case of two boys born on Wednesday need not be included in our enumeration of cases because it wasn't possible for both boys to be born on Wednesday, since we know one was born on Tuesday! I hope everyone can see the post facto fallacy in this reasoning. Anyone who would buy this line of reasoning is playing the role of the father in the following comic scenario:

One day you get a letter from the town correcting your son's birth certificate. He was born two seconds after midnight so he was actually born on a Wednesday rather than a Tuesday. With a heavy heart, you sit your son down and tell him the unfortunate news: "I'm sorry to tell you this son, but I'm not your real father. My son could only have been born on a Tuesday, and I've just learned you were born on a Wednesday."

By the way, this problem is not comparable to the Monty Hall problem. The Monty Hall problem is a genuinely counter-intuitive probability result. The Tuesday's Child problem is more like a riddle or joke that depends on deceptive or ambiguous language.