Originally Posted by

**NORM** GordonX, you have a case of justifiable confusion. Let's start from the beginning. Suppose someone tells you that one of a pair of kittens is a boy and the other is unknown (as in the book Paradox, pages 18-19) and you want to know the odds that both are boys. There are four equally likely possibilities: BOY-BOY, BOY-GIRL, GIRL-BOY, GIRL-GIRL. This maps to the case of the four equally likely possibilities when you flip two coins: 1/4 chance for HEADS-HEADS, 1/4 chance for HEADS-TAILS, 1/4 chance for TAILS-HEADS, 1/4 chance for TAILS-TAILS. If you learn that one kitten is male, it’s like learning that one coin toss landed heads. (I imagine coins more readily than kittens. Please bear with me.)

Now suppose a machine (or a friend) flips two coins, randomly picks one of the coins and tells you whether it’s heads or tails. Let’s say that it’s heads. According to Paradox, you should now think that there is a one in three chance that both coin tosses landed heads and a two in three chance that one is heads and the other tails. Similarly, if the machine says tails, you should now think that there is a one in three chance that both coin tosses landed tails and a two in three chance that one is heads and the other tails. Let us add the probabilities, according to Paradox, of when the machine says heads and when it says tails. When the machine says heads, this yields odds of: 1/3 heads and 2/3 heads + tails (in either order). When the machine says tails, this yields odds of: 1/3 tails and 2/3 heads + tails (in either order). Half the time the machine will say heads and half the time it will say tails, so we must multiply each set of probabilities by a half and then add them together. This yields: 1/6 heads; 2/3 heads + tails (in either order); 1/6 tails. This fails to match the odds given in the first paragraph, so Paradox is wrong.

When our machine announces heads, it has picked one of four (not three) heads in the possibilities list: two in HEADS-HEADS, one in HEADS-TAILS and one in TAILS-HEADS. We may list the possibilities as: HEADS-HEADS, the other HEADS-HEADS, HEADS-TAILS, TAILS-HEADS. The odds that the unknown coin toss will be heads is 1/2. Similarly, when the machine announces tails, it has picked one of four (not three) tails in the possibilities list. When we sum these probabilities, they match those in the first paragraph.

An intuitive approach supports this. Flip two coins (these may be black, tabby, or whatever), let them land on a table, pick one at random, announce whether it is heads or tails, set it aside and intuit the odds of the second coin being heads or tails. See why it’s easier to think in terms of coins? (It’s harder to flip cat genitals.)

I emailed Jim Al-Khalili, the author of Paradox, but he didn’t respond.