Defeating Derren Brown

In one Derren Brown show, “Inside Your Mind“, he plays his “favorite game” with 3 Psychology students. He puts money in 1 of 2 envelopes, and the 3 Psychology students have to guess which envelope contains the money. If they guess right, they get to keep the money.

2 envelopes

Derren first plays with 5 pounds and shuffles while the 3 Psychology students face the other way. When they look back, he says some things and makes certain gestures in an attempt to influence their choice of envelope. They pick and, of course, lose.

The second round is for 10 pounds, and Derren shuffles under the table. He again says some things and makes certain gestures in an attempt to influence their choice of envelope. Again, they pick and, of course, lose.

The third round is for 20 pounds, and Derren shuffles under the table. When he puts them on the table, he pushes one envelope towards them. He again says some things and makes certain gestures in an attempt to influence their choice of envelope. Again, they pick and, of course, lose.

The fourth round is for 50 pounds. This time the Psychology students hide the 50 pounds in 1 of 3 envelopes, and Derren has to guess which envelope contains the money. They, of course, lose.

The reason why they lost every time is because their strategy is to pick whatever they feel like picking, which is wrong since what they feel like picking is influenced by a mentalist.

Their strategy has an expected value of  \Sigma _{-\infty}^{\infty} xp(x) = (5)(\frac{0}{2}) + (10)(\frac{0}{2}) + (20)(\frac{0}{2}) + (50)(\frac{0}{3}) = 0.

Here is an alternative strategy based on the first arc of the second series of Yu-Gi-Oh!:

In Yu-Gi-Oh!, the main characters, Yugi and Atem, share a body and play a children’s card game hoping to save Yugi’s grandfather from the arc villain Maximillion Pegasus, who has the ability to read minds. In the first part of the game, Atem plays. Pegasus, being a mind reader, obviously counters all of Atem’s strategies. Eventually, they come up with the idea of Yugi playing instead of Atem and then switch to Atem whenever Pegasus tries to read their mind. Thus, they are able to successfully hide their ace card, Dark Magician, in Magical Hats and defeat Pegasus’ Toon Monsters.

The application: Get Derren to be unable to read your mind/influence by choosing an envelope randomly.

This new strategy has an expected value of \Sigma _{-\infty}^{\infty} yp(y) = (5)(\frac{1}{2}) + (10)(\frac{1}{2}) + (20)(\frac{1}{2}) + (50)(\frac{2}{3}) > 0 .