Double-Six Puzzle

The Double-Six puzzle is a variation of the Two Child Problem. I wrote it to demonstrate that many classic and counterintuitive probability problems are ambiguous and therefore unanswerable.

Puzzle 1

You roll two dice. One of them falls under the table and you can’t see it. The other one lands on top of the table, and it’s a 6. What is the probability that both dice landed on a 6?

Puzzle 2

You are rolling two dice blindfolded. A machine is programmed to ding if and only if at least one of them lands on a 6. You keep rolling until the machine dings. What is the probability that both dice landed on a 6?

Solutions

Puzzle 1: The probability that both dice landed on a six are 1 in 6.

Puzzle 2: The probability that both dice landed on a six is 1 in 11.

Why the difference?

In puzzle 1, the die under the table is a complete mystery, so it still has its usual chances: one chance in six of also being a 6. Seeing a 6 on a particular die doesn’t change the second die’s ordinary roll chances.

In puzzle 2, all you know is that somewhere in the pair at least one 6 has appeared. If you look at all 36 outcomes of a roll of 2 dice, you see there are 11 different outcomes in which there is at least one 6. Only one of those 11 outcomes is the double six.

The Lesson

Imagine I had phrased the question like this:

"You roll two dice. At least one lands on a six. What are the chances that both landed on six?"

This doesn't tell you whether you're in a situation more like puzzle 1 or puzzle 2, so you wouldn't have been able to answer it.

The Two Child Problem (also called the Boy or Girl Paradox) is typically framed like this:

"Mr. Smith has two children, at least one of whom is a boy. What are the chances that both his children are boys?"

This phrasing doesn't tell us how we know that Mr. Smith has at least one boy, so the answer could be 1/2 or 1/3 (assuming a 50% chance of having a son with each birth).

In many probability problems, how we came to know the information matters. Without specifying, the questions are ambiguous and, therefore, unanswerable.