Re. the Roy Meadows/Sally Clark example, you say:

a pediatrician had testified that the odds of two children in the same family dying of infant death syndrome was 73 million to 1. Unfortunately, he had arrived to this figure by squaring the odds of a single death. Squaring the odds of a single event to arrive at the odds for it happening twice only works if the two events are independent. But that assumption is likely to be false in the case of multiple deaths in the same family

More importantly, the 73 million to 1 number was completely irrelevant.

The interesting number is not "the odds of two children in the same family dying of infant death syndrome" but "the odds of two children in the same family having died of infant death syndrome given that two children in that family had died", which are obviously much higher.

Of course, you need to know the first number in order to calculate the second (using Bayes' theorem) but Meadows (and everyone else present at the trial) managed to conflate the two.

edit 05/07: corrected bizarre thinko at the end of penultimate paragraph.