This sort of problem is often easier to understand when modified to make the probabilities more different. E.g. suppose ten children and information that at least nine of them are boys. The incorrect reasoning leads to 1/2 probability of ten boys, while actually the probability is only 1/11. You can even write a program which generates a sequence of ten binary values, 0 for a boy and 1 for a girl, and then prompts you whenever it encounters at least nine zeros and compare the relative frequencies. If the generated binary numbers are converted to decimals, it means that you generate an integer between 0 and 1023, and get prompted whenever the number is a power of 2, which correspond to 9 boys and 1 girl (10 possible cases), or zero, which corresponds to 10 boys (1 case only).

Such modification works well as an intuition pump in case of the Monty Hall problem, maybe is not so illustrative here. But Monty Hall is isomorphic to this one.