Three Card Monty Hall

Imagine we are playing a game of cards. In this game, there are only three cards in the deck: An Ace and two Kings. I will deal you one card, and I will keep the other two. You win if, at the end of the round, you are holding the Ace. You are not allowed to look at your card, but you’re allowed to trade hands with me. Do you?

As it stands, this is an easy problem: Of course you do. There’s a 1/3 chance you have the Ace and a 2/3 chance I have the Ace. I either hold two Kings (and you hold the Ace) or I hold the Ace and a King, so we both already know that I have at least one King. I look at my cards, and show you that I do indeed have at least one King. After showing you, I make the offer: Do you want to trade hands?

At this point, a strange thing happens for a lot of people: Suddenly, trading doesn’t seem as tantalizing. Even people who understand the problem think that my showing you the King counts as new information. When I tried this experiment with my wife, who understands the Monty Hall problem just fine, she also hesitated to trade once she’d seen the King.

Again: We both knew at the time of the deal that I have at least one King. That’s a logical requirement. Because I have two cards, and because there is only one non-King in the deck, it’s simply impossible for me not to have a King. So there’s no new or interesting information in my revealing this fact by showing you a King. I’m telling you absolutely nothing you didn’t already know.

Why, then, is it less tempting to switch hands now?

One thought I had comes from my wife’s reaction to seeing the King. I’ve actively shown you that I have a losing item. It’s one thing to know this logically, but another thing to actually see it. I’m a partial loser.

There’s also the illusion that my reveal somehow changed the odds. If I had randomly shown you one of my two cards, there’s a 1/3 chance I’ll show you the Ace (in which case you lose) and a 2/3 chance I’ll show you a King. In that case, you would have a 1/2 chance now, but that’s because of the 1/3 chance you had of losing with my first reveal. These are the scenarios if I randomly show you one of my cards:

  1. You have K (2/3), I have A and K, I show A (2/3 * 1/2 = 1/3)
  2. You have K (2/3), I have A and K, I show K (2/3 * 1/2 = 1/3)
  3. You have A (1/3), I have two K, I show K (1/3 * 2/2 = 1/3)

If I show you a K, you know it must be either of the last two scenarios, which have an equal chance of occurring. But the first scenario is barred by the rules of the game: It is the rules of the game, not the reveal, that determines the odds. I cannot reiterate enough: You already knew I had at least one King, so my showing you one shouldn’t change your strategy one iota.

The actual Monty Hall problem has other confounding features as well. Technically, the offer is not to switch a one card hand with a two card hand but rather to switch one door for another. But that makes no difference to the probability, because Monty Hall could let you have the revealed goat too.

In an earlier item, I pointed out that the real Monty Hall didn’t play by the rules of the puzzle on the real Let’s Make a Deal. This is something else that even people who understand the problem get wrong.

And it’s also been suggested that some of the controversy may be men with degrees scoffing at a woman without one. Given the nature of the times, and the tenor of some of the comments shown in that article, that’s certainly possible.

Furthermore, there are human factors involved in gambling, such as our predisposition to think we have “good luck” or “bad luck” when making random choices.

Regardless, hopefully seeing the Monty Hall problem in terms of the three-card game above provides a fresh perspective for some people.

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