Gambling and Expected Value: Three-Card Monte

In this post on Gambling and Expected Value, we look at Three-Card Monte.
Click here to find similar posts on other lotteries and games of chance.

How the Game Works

The game Three-Card Monte is typically just a con to sucker anyone dumb enough to play out of all their money, but we're going to assume for this blog post that we're dealing with a completely fair version of the game. A dealer has three cards on a table, a "money card" and two others. The money card is typically the Queen of Hearts and the other cards are typically the Jack of Spades and Jack of Clubs. A player is asked to bet that they can find the money card after the dealer has turned the cards face down and expertly re-arranged them. The player guesses which card is the Queen and wins if he guesses correctly. The wager is lost if the player is wrong. 

Probabilities and Prizes

The player is generally led to believe that if he watches closely, he can follow the Queen at all times and know where it ends up when the dealer is done shuffling. However, during shuffling a skilled dealer stacks up two cards in one hand and can throw either the bottom card or the top card down at will. When executed correctly, the player can't tell which of the two cards was thrown. Therefore, it becomes a game of chance for the player rather than a game of skill. There is a 1 in 3 chance of revealing the money card, which pays out 1:1 on the wager.

Expected Value

The expected value of a fair version of Three-Card Monte is easy to calculate because there is only one way to play and only two possible outcomes. There's a 1 in 3 chance of winning the wager and 2 in 3 chance of losing it. Therefore, the expected value is -1/3 times the wager. In other words, for every dollar wagered, on average, you're losing a third of it.

So playing Three-Card Monte is a good way to throw money away even if the game is played fairly. However, if played fairly, Three-Card Monte is a smarter gamble than 50/50 draws or lotteries. That said, keep in mind this analysis was merely hypothetical. Three-Card Monte is a con game and a number of tricks are used to ensure that the player never wins a single bet. In other words, your true expected value in a realistic game is -1, or at least much closer to -1 than my hypothetical analysis suggests.