In William Shakespeare’s ‘Merchant of Venice’, potential suitors of young Portia are offered a choice of three caskets, one gold, one silver and one lead. Inside one of them is a miniature portrait of her. Portia knows it is in the lead casket.

Now let us think about a plot twist where Portia must open one of the other caskets and give Arragon a chance to switch choice of caskets if he wishes. She is not allowed to indicate where the portrait is and in this case must open the gold casket (she knows it is in the lead casket so can’t open that) and show it is not in there. She now asks the Prince whether he wants to stick with his original choice of the silver casket or switch to the lead casket.

So should he switch to the lead casket or stay with the silver? It depends whether things actually are equal. In particular, it depends on how valuable any information contained in the inscriptions is. If he has little faith in the inscriptions to arbitrate, he should definitely switch and improve his chance of winning fair Portia’s hand from 1/3 to 2/3. If he thinks, however, that he has unlocked the secret from the inscriptions, the decision is more difficult. If so, he might stick with his choice in good conscience.

In summary, the key to the problem is the new information Portia introduced by opening a casket which she knew did not contain the portrait. By acting on this new information, the Prince can potentially improve his chance of correctly predicting which casket will reveal the portrait from 1 in 3 to 2 in 3 – by switching boxes when given the chance. Unless he has other information which makes the opening probabilities different to 1/3 for each casket, such as those cryptic inscriptions. If this information is potentially valuable, or at least if the Prince thinks so, that complicates matters!

Exercise