confused about published answer to a conditional probability question

Solution 1:

The answer is correct: If we know Bob bought a green bike, then there are only three other green bikes that Alice could have bought out of the eight bikes Bob didn't buy. Thus the probability that Alice bought a green bike given we know Bob bought a green bike is $3/8$.

It is not that Bob's subsequent choice is affecting what Alice did in the past. It is that knowing Bob's choice gives us information about what Alice did in the past. The fact that Bob also happened to buy a green bike, when choosing randomly, makes it less likely that Alice had (because, symmetrically, when Alice buys a green bike, it is less likely that Bob would buy a green bike randomly).

Note that $P(B|A)=P(A|B)$ here. The order Alice and Bob buy bikes does not matter.

A simpler example to illustrate the point: Suppose that the bike store had one green bike and one red bike. Then:

• if you know Bob bought a green bike, it tells you Anne could not have bought a green bike: $P(A|B)=0$;
• if you know Alice bought a green bike, it tells you Bob did not buy a green bike: $P(B|A)=0$.