Our predilection for causal thinking exposes us to serious mistakes in evaluating the randomness of truly random events. For an example, take the sex of six babies born in sequence at a hospital. The sequence of boys and girls is obviously random; the events are independent of each other, the number of boys and girls who were born in the hospital in the last few hours has no effect whatsoever on the sex of the next baby. Now consider three possible sequences:

BBBGGG

GGGGGG

BGBBGB

Are the sequences equally likely? The intuitive answer- "of course not!" -is false. Because the events are independent and because the outcomes B and G are (approximately) equally likely, then any possible sequence of six births is as likely as any other. Even now that you know this conclusion is true, it remains counterintuitive, because only the third sequence appears random. As expected, BGBBGB is judged much more likely than the other two sequences.

I think he is confusing two popular fallacies with a statistical fact.

Fallacy 1. After six girls, a boy is more likely (or indeed, after 666 girls). This is obviously false.

Fallacy 2. BBGBGG is more random/more likely than GGGBBB or GBGBGB. This is less obviously false.

Fact. The sexes in a run of six are most likely to be 3G3B (in any order), then 2G4B or 2B4G, then 1G5B or 1B5G, with six girls or six boys the least likely. (The first child must be one sex - with one in 2000 intersexed, but we'll ignore that - it's one chance in two the next child will be the same sex, one in four there will be three in a row, one in 8 four in a row, and so on.) So in any run of six, the chances of all one sex are one in 2^5 or 32, and the chance of that sex being girls is half again, so the chance of GGGGGG is one in 2^6 or 64. Expressed differently, the chance of G is one in 2, of GG one in 4 (with equal chances of GB BG or BB) GGG one in 8 (vs GGB, GBB, GBG, BBB, BBG, BGG and BGB) etc.

So has Kahneman overstated his case, or have I missed something?