Let me go through this scenario by scenario.

The key assumptions are (1) the predictor’s prediction is not influenced by your actual choice, and (2) it is not changed once made.

First scenario: The predictor has been correct 90% of the time, but that figure does not reflect the probability of it being correct for this trial.

Suppose the predictor predicts the one box. If you take the one, you will get 1 million. If you take both, you will get 1 million 1 thousand.

Suppose the predictor predicts both boxes. If you take the one, you will get zero. If you take both, you will get 1 thousand.

Therefore, your choice determines your expected value, and the highest expected value comes from taking both boxes.Second scenario: The predictor has a 100% probability of being correct for this trial.

Suppose the predictor predicts the one box. If you take the one, you will get one million. However, because of the key assumptions, it would beimpossiblefor you to take both, as that would mean the predictor predicted wrong.

Suppose the predictor predicts both boxes. If you take both, you will get 1 thousand. For the same reasons as before, it would be impossible for you to pick only one in this case.

Therefore, your expected value is completely dependent on what the predictor predicts. You have no say in the matter.Third scenario: The predictor has a 90% probability of being correct for this trial.

Suppose the predictor predicts the one box. Then you have a 90% chance of taking the one for 1 million, and a 10% chance of taking both for 1 million 1 thousand, with an EV of 1 million 1 hundred.

Suppose the predictor predicts both boxes. Then you have a 90% chance of taking both for 1 thousand, and a 10% chance of taking the one for 0, with an EV of 9 hundred.

Your EV is once again dependent on what the predictor chooses.Fourth scenario: You are given no information whatsoever on the reliability of the predictor.

Either the predictor determines your EV, or you do. In all the cases where you determine your EV, it is better to take both boxes.

Therefore, you should try to take both boxes just in case you actually are in control. If you are not, then you would be controlled by probability and not actually have a choice.

Thanks Mingy Jongo for going through all these scenarios.

I’ll be back once I’ve had time to think.

Stephen