Open the Box or Take the Money?

“Would you like to open the box or take the money”? This was the standard line of the original ‘quiz inquisitor’, Michael Miles, on his show ‘Take Your Pick’, a popular favourite in the early days of commercial television. The idea was to present contestants with a choice between opening a box, which might contain anything from a car to a rotten tomato, and receiving a wad of notes in the hand. Many, but not most, chose the money. Now let’s change the format of the game a little and offer contestants a choice between three boxes, one of which contains a cheque for a million pounds and the other two which are empty. Let’s call the boxes “Gold, Silver and Lead”, the choice of caskets offered to Portia’s suitors in Shakespeare’s ‘The Merchant of Venice’. Let’s say that, like Bassanio, you choose the box made of lead. I am the quiz inquisitor on this occasion and I now open the box made of gold. It is empty. At this point I offer you the opportunity to stick with your original choice or to switch. What should you do, and does it matter? Basic intuition may tell you that there are two remaining boxes, of silver and lead, and so it should be an even chance of winning whichever of these boxes you select. If this line of reasoning is correct, it makes no difference to your probability of winning the prize whether you switch to the silver box or stay with your original choice of lead? But would this intuition be correct? To answer this we need to ask one simple question. Has any new information been introduced by my decision to open the gold box? This depends on whether I know which box contains the cheque when I open the box. To take an example, assume I know that the box with the prize is the silver box. When you choose the lead box, I now have no choice but to open the gold box. In effect, then, I am actually pointing out to you which box contains the prize. This is the case whenever the box you have chosen is empty. There is a 2 in 3 chance of this as there are two empty boxes and only one containing the prize. There is a 1 in 3 chance that the box you have chosen is in fact the box containing the cheque. In this case, it doesn’t matter which box I as the quizmaster choose to open as they are both empty. To summarize this, there is a 2 in 3 chance that I am pointing out to you the winning box. It is the box which I chose not to open, in this example the silver box. There is a 1 in 3 chance that you chose the right box in the first place. So what’s your optimal strategy? This becomes clear once you realize that you only have a 1 in 3 chance of winning if you stick with your original box but a 2 in 3 chance if you switch to the box which I didn’t open. The key to the riddle is the new information I introduced by opening the box which I knew to be empty. By acting on this new information, you can improve your chance of correctly predicting which box will open to reveal the cheque from 1 in 3 to 2 in 3 – by switching boxes when given the chance.