Question

There are four medals (Gold, Silver, Bronze and Wood) on a table, but they are all wrapped with dark wrapping paper, such that it is impossible to distinguish them. You would like to find the gold medal.

The game starts as follows. You pick one medal without unwrapping it, and then the game host unwraps one of the remaining medals and reveals that it is a silver medal. (Assume here that the host unwraps a medal with equal probability, but knowing where the gold medal was and avoiding unwrapping the gold medal if still on the table, to keep the game interesting to watch until the end.)

You have now three medals left to unwrap (one in your hand, two on the table). At this point, the host gives you the option to change your mind and swap your medal for one of the two left on the table.

What would you do at this point? Would you keep your medal, or swap it with one of the two medals left on the table? If so, which one?

Hints: Find the solution by using Bayes’ theorem, calculating all the conditional probabilities involved. Start calculating the probability of having Gold in our hands

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given that we know that the host unwraps Silver, 𝑃(𝐺|𝐻𝑠) = . . ..

Then compare with the probability of having Bronze or Wood in our hands given that we know that the host unwraps Silver, 𝑃(𝐵|𝐻𝑠) = . . .., 𝑃(𝑊|𝐻𝑠) = . . ..