Is it possible that individually, pieces of evidence increase the chance of guilt but together they decrease the chance of guilt?

Question

Let G be the event that a certain individual is guilty of a certain robbery. In gathering evidence, it is learned that an event $E_1$ occurred, and a little later it is also learned that another event $E_2$ also occurred. Is it possible that individually, these pieces of evidence increase the chance of guilt (so $P(G|E1)$ $> P(G)$ and $P(G|E2) > P(G)$), but together they decrease the chance of guilt (so $P(G|E1, E2)$ $ < P(G)$)?

This is the example in the solution:

Suppose that the crime was committed between 1 pm and 3 pm on a certain day. Let E1 be the event that the suspect was at a nearby coffeeshop from 1 pm to 2 pm that day, and let E2 be the event that the suspect was at the nearby coffeeshop from 2 pm to 3 pm that day. Then P(G|E1) > P(G), P(G|E2) > P(G) (assuming that being in the vicinity helps show that the suspect had the opportunity to commit the crime), yet P(G|E1, E2) < P(G) (as being in the coffeeshop from 1 pm to 3 pm gives the suspect an alibi for the full time).

I've been trying to think of another example but I can't think of any. Can someone provide another example? Moreover, in terms of general problem-solving heuristics, I find myself terrible at coming up with examples/counter-examples. My only strategy is to look for simple/extreme cases, but there's no simple/extreme cases for this question.

Thank you very much!

Answer 1

Assume that Person-1 only wears the shoes that he owns, and that he only wears a left shoe on his left foot and a right shoe on his right foot.

Assume that both tangerine shoes and red-polka-dot shoes are very unusual. Assume that Person-1 owns both a left tangerine shoe and a left red-polka-dot shoe, but does not own any right shoes that are tangerine or red-polka-dot.

One witness says that at least one of the shoes worn by the perpetrator was tangerine, but the witness unsure whether shoe was on left foot or right foot.

2nd witness says the same, except with respect to red-polka-dot shoe. So police conclude that perpetrator owns a tangerine-left and red-polka-dot-right pair of shoes, or vice-versa.

Each witness individually casts legitimate suspicion on Person-1, but collectively, they exonerate him.

Answer 2

Here is an example based on "consumable" items. Suppose that a suspect has only enough money/resources to buy one expensive lockpick, and it can be shown by looking at his bank account that he only made one purchase of such high-level lockpick. Furthermore, this lock pick has come out so recently that you cannot find it anywhere- no one drops it, and no one lends it to others for free. This lock pick, being designed to pick the most difficult-to-overcome locks, disintegrates upon use. Suppose that it was found that a lock was picked at the scene of the crime that required a lock-pick of that caliber to get through. Later, it was found that another lock in the same building was picked too. Individually, the pieces of evidence show that the man could certainly have been the thief because he purchased a lock pick that could get through those locks. But together, he could not have been the one because he did not have two lock picks.