Introduction

Humans don’t only use shortcuts to come up with quick estimates, as we discussed in the previous chapter. We also try to get away with them for more complex problems! We’ll explore humans’ difficulties with two specific problems in this chapter: the Wason Card Task and the Monty Hall Problem.

Logic and the Wason Card Task

Do you like logic problems?

Let me clarify. Do you like problems that go, “if x is true and y is not true, is z sometimes true?” Some people enjoy these considerably, while other people loathe them with every fiber of their being. What does everyone have in common, though? None of our brains are wired to compute these automatically and accurately (computers, on the other hand, are wired exactly right to compute these automatically). You can teach yourself to consciously solve these problems, but knowing the answer to a logic problem is not like knowing what a chair is. Let’s illustrate this with a problem following the pattern I outlined above—an if-then question known as the Wason Card Selection Task¹.

Suppose you have a stack of cards. Each card has a letter on one side and a number on the back. There are four cards laid out before you, like in Figure 9.1. The cards are supposed to follow a rule: if a card has an even number, then it must have a vowel behind it.

Your task is to examine these cards and make sure that all of them are following the rule – efficiently! Theoretically, of course, you could just flip over every card and see what’s on the other side. But let’s assume you have many, many more of these to check, and if you can figure out how to only check the ones you absolutely have to check, that would be best. So, consider the rule:

If a card has an even number on it,

Then it must have a vowel on the other side.

And determine: which card(s) do you absolutely have to flip over in order to make sure all cards are following the rule?

Give it a try! You may have a few ideas. A common thought process is similar to Liz’s below, in the upper panel of Figure 9.2.

If you were thinking similar to Liz, or you selected either the 14 card or the A card to check, you would be like most people! In this case, it’s not the correct answer – but remember, wrong answers in these sorts of problems don’t mean you don’t have a well-functioning brain. Humans are not naturally good at this kind of computation. The correct solution is illustrated in the lower panel of Figure 9.2.

Did you catch that you need to check the B in addition to the 14? Did you also catch that you don’t need to check the A at all? Let’s briefly go through why Agatha’s solution is the correct one. In a logic problem set up like if X, then Y, you should definitely check all instances of X, to make sure that Y goes with them. In this case, you’d be checking even numbers, to make sure there are vowels behind them. But what many people forget about is that every if X, then Y statement comes with a secret, implied if NOT Y, then NOT X statement. Confused? It might help to substitute in some words. Consider this:

If X, then Y

If Tolly is a rat, then he is an animal.

If NOT Y, then NOT X

If he is NOT an animal, then Tolly is NOT a rat

Both statements about Tolly are true. In fact, in these kinds of logical statements, if the if X, then Y statement is true, then the if NOT Y, then NOT X statement is also always true! They go together. So, whenever we evaluate an if X, then Y claim, we always have to remember to check the if NOT Y, then NOT X claim.

Before we leave logic (to go on to everyone’s other favorite topic, probability) I would like to pose one more logic problem. Suppose you are a bouncer at a bar where the law is that no one under the age of 21 may drink alcohol. Your job is to make sure that no one is breaking the law! What you see is detailed in Figure 9.4: an adult, a child, someone who ordered an apple juice, and someone who ordered a beer. Who do you absolutely need to check to make sure they are not breaking the law?

If you and Sunny had the same thought process, this time, you’d be absolutely right! In fact, more people get this kind of question right². You might even find it downright easy. But, as observant readers may have noted, this is the exact same problem as the Wason Card Task. Both questions boil down to “The rule is if X, then Y. Do you need to check X? (Yes) Do you need to check Y? (no) Do you need to check NOT-X? (no). Do you need to check NOT-Y? (yes).” In fact, you can even more directly substitute the problems for each other: in the Wason Task presented earlier, the numbers are 51 and 14. You can think of these as ages. The letters are A and B. You can think of these as apple juice and beer.

Why is the “alcohol rule check” problem so much easier for people than the original Wason Card Task? There are many proposed explanations, but a broad one we’ll go with here is simply context. Context is the magical glue that makes meaning of things; it allows our brains to stop saying “how do I solve this logic problem” and translating it to something easy and familiar. We’re not used to consciously stating if X then Y, but we are used to thinking about what we’re allowed to do and when we’re allowed to do it. The context of the bar allows us to skip the part where we try to consciously work out something our neurons have already figured out how to do.

Probability and The Monty Hall Problem

One of the most famous problems in decision science is called the “Monty Hall Problem.” It gets its name from a game show host (Monty Hall) and the show “Let’s Make a Deal,” which started in the 1960s. Here’s the scenario:

You’re looking at three closed doors. You know that behind one of the doors is a brand-new car! But behind the other two doors is…a goat. Assume that you don’t want a goat.

Monty Hall asks you to select a door. Suppose you select Door #1.

“Okay,” says Monty Hall, “You’ve selected Door #1. But before I show you what’s behind Door #1, I want to show you one of the doors you didn’t pick!” Suppose Monty Hall then opens Door #3, and reveals a goat behind it.

“Now,” he says, “You have a choice. Do you want to stay with Door #1, or do you want to switch to Door #2?”

You can examine this scenario as illustrated in Figure 9.6.

Before we resolve the scenario, we should stop to consider the question of switching. Should you switch from Door #1 to Door #2? Should you stay with Door #1? Does it even matter?

Many people have a gut feeling that it should not matter whether you switch or stay. When you make your initial choice, you have a ⅓ chance of landing on the car. Opening one of the doors doesn’t go back in time and change the probability of that initial choice. So, in our minds, the probability of either Door #1 or Door #2 having the car is still ⅓:

But this is wrong! In fact, you are more likely to get the car if you switch. In fact, you are twice as likely! The “correct” answer to the Monty Hall Problem is to always switch your initial choice.

Does that feel right? If it doesn’t, don’t worry. It is true that when you make your initial selection, the odds of you getting the car are one in three: ⅓. Learning the contents of one of the doors you didn’t pick doesn’t change that. However, it does change the probability of the other door! When you make your initial choice, everything is random. But once Monty Hall chooses to show you a goat, that is not a random choice. Let’s look at it this way, with every possibility laid out:

• If Door #1 contains the car, Monty will show you one of goats behind one of the other doors.
• If you STAY, you will get the car!
• If you SWITCH, you will get a goat.
• If Door #2 contains the car, Monty will show you the goat behind Door #3
• If you STAY, you will get a goat.
• If you SWITCH, you will get the car!
• If Door #3 contains the car, Monty will show you the goat behind Door #2
• If you STAY, you will get a goat.
• If you SWITCH, you will get the car!

One out of three times, you will have chosen the car at random. Two out of three times, you will have chosen one of the goats.

In other words:

• You’d only get the car by switching if you picked a goat initially
• You initially pick a goat ⅔ of the time!

If this has sunk in for you, fantastic! If you are like me though, you might see the all the scenarios laid out, see the math, and say “yes, I understand why this is true,” but maybe not be able to really feel it in your gut. If you feel this way, you are not alone. This is why the Monty Hall Problem is famous: the solution just doesn’t feel right.

Chapter 9 Wrap-Up

Both problems we discussed are examples of how humans naturally struggle when directly faced with logic or probability. This does not mean we are stupid; nor does it mean that you are abnormal if you get good at these things. Rather, all it means is that we are not computers. Computers are quite literally built on logic. Our organic, squishy brains, on the other hand, are built on organic, squishy chaos. We learn better through experience and association. If you did the Monty Hall Problem a hundred times, it would probably start to sink in that switching is better than staying. (Can we link to a site that does that? Or is that not OER?)