It’s an interesting tool ]]>

Q1) Obviously, not all can ask it about Tuesday Boys, as Gary Foshee did. In fact, only 27 can. But 26 of those could ask a different question. How many do you expect to?

Q2) Is your expected number different for any description of a child, such as “Girl born on a Thursday?” If so, why?

Q3) How many of those, whom you expect to ask about have Tuesday Boys, have two boys?

Q4) How many of those who ask the question any about any specific description have two of the same gender?

Q5) What is the best answer to Gary Foshee’s question?

Answers, in a slightly different order:

A2) You have no information that would allow you to expect the number of questions to be different for different descriptions. Since there are 14 descriptions, and 196 questions, you should expect 196/14=14 questions about each description.

A1) 14. Not 27.

A4) You have no information that would allow you to expect the fraction of same-gender families to be different for different questions. Since half of the entire group has same-gender families, you should expect 1/2 of each group to have same-gender families also.

A3) 7 of 14. Not 13 of 27.

A5) 1/2.

]]>Q1) You pick a box at random. What are the chances it contains only one type of coin? This supposed to be easy: 1/2.

Q2) Say I peek in the box, and tell you that there is (at least) one bronze coin inside. What is the probability that there are two? Note that this question is now identical to Robin Ryder’s “well-known probability riddle.” He says the answer changes, to 1/3.

Q3) What if I had said that there is (at least) one gold coin in the box. Shouldn’t the probability change to 1/3 in this case, also?

Q4) What if, instead of naming a type, I reach in, and pull a out a coin in my closed fist. Now what is the probability that the coin still in the box is the same type?That is, instead of naming “bronze” in Q2, I would grab a bronze coin, and instead of naming “gold” in Q3, I would grab a gold coin. So this question is the same as Q2 adn Q3, and Robin Ryder said the answer is 1/3.

It seems that my action in Q4 has to change the probability no matter what I picked. But such a change can only happen if useful information is gained, and it has not. This is the paradox that Bertrand referred to.

The resolution of the paradox is that naming a type of coin does not allow you to infer that I was forced to name that type. If the box did, indeed, have two of the same type, then I was forced. But if it had both types, I had to choose, and you can only assume I chose randomly. That makes the answer to Q1, and Robin Ryder’s riddle, 1/2.

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