Probability, Why Intuition Fails Us & How Design And Simulation Can Help.
Goats, cars, doors.
Intuition often seems to fail us when dealing with probabilities. We have strange cognitive biases that have important implications for the way we see the world, make choices, and design things. There are several ways in which probability doesn’t make sense, and yet it’s such an innately useful tool.
One of the most famous examples of how our intuition fails us is the Monty Hall problem. I re-encountered this problem the other day through this great online MIT course I’m taking: Great Ideas in Computer Science (lecture on Randomness). The prof references the problem as an example of the “Slippery Nature of Probabilities”. Here’s his explanation of Monty Hall, that demonstrates how our probabilistic intuitions tend to be misguided:
Suppose you are on a game show. There are three doors, of which one has a car behind it, and the other two have goats. The host asks you to pick a door; say you pick Door #1. The host then opens another door, say Door #3, and reveals a goat.
The host then asks you if you want to stay with Door #1 or switch to Door #2. What should you do?
In probability theory, you always have to be careful about unstated assumptions. Most people say it doesn’t matter if you switch or not, since Doors #1 and #2 are equally likely to have the car behind them. But at least in the usual interpretation of the problem, this answer is not correct.
The crucial question is whether the host knows where the car is, and whether (using that knowledge) he always selects a door with a goat. It seems reasonable to assume that this is the host’s behavior. And in that case, switching doors increases the probability of finding the car from 1/3 to 2/3.
To illustrate, suppose the car is behind Door #1 and there are goats behind Doors #2 and #3.
Scenario 1: You choose Door #1. The host reveals a goat behind either Door #2 or Door #3. You switch to the other door and get a goat.
Scenario 2: You choose Door #2. The host reveals a goat behind Door #3. You switch to Door #1 and get the car.
Scenario 3: You choose Door #3. The host reveals a goat behind Door #2. You switch to Door #1 and get the car.
From Wikipedia:
The whole thing is a bit of a mind-fuck, yet if you do the math, it seems to work out.
N-Doors
Professor Aaronson goes on to give the following explanation that often tends to help people understand the intuition behind what’s happening:
Here’s another way of thinking about this problem. Suppose there are 100 doors, and the host reveals goats behind 98 of the doors. Should you switch? Of course! The host basically just told you where the car is.
Having taken a bunch of Stats courses while studying Economics, I’ve encountered the Monty Hall problem several times across the years. I already know the mathematical solution, yet every time I hear the 3 door example, the notion that you should switch after the host opens a door continues to go against the grain of my intuition — it causes me discomfort. I’m clearly not the only one: apparently Paul Erdos, one of the most famous mathematicians of all time, refused to believe switching was beneficial until someone finally showed him a computer simulation of the results. The Wikipedia article on the Monty Hall Problem really shows how unintuitive this problem is for most. The anecdote about Paul Erdos, also pushes us into an interesting corollary exploration about computer simulations convincing us of truths and probabilities.
I think there’s something significant in the last little tidbit about the problem with 100 doors instead of 3. The way I always mend my intuition is to consider the 100 door example. When there are 100 doors, as opposed to only 3, it immediately becomes obvious that if the host opens 98 of the doors, she is giving you a lot of information about where the goat is likely to be. Simply put, it’s clearly unlikely that you choose the door with the goat on your first try (1/100 odds), so if the host is forced to eliminate 98 of the doors and leave you with only your original door, and another door, it’s very likely that the goat is behind the other door that the host has been forced to keep rather than behind the originally chosen door. It seems that people in general are better able to understand the intuition behind switching when the set of doors is large.
So what’s going on here?
When N is large (100 doors), we’re able to see the informational impact of the host’s action clearly — we feel the magnitude of the information they’ve revealed through the action of opening a door, and have an intuitive sense that our odds of picking the door with the goat will greatly improve when switching (1/100 to 99/100, 99x improvement). In the 3 door example, our odds don’t improve as much (1/3 to 2/3 , 2x improvement), and our intuition on the smaller set seems to fail us. When the improvement in odds is minor, we don’t immediately see that the host’s action has revealed new information to us, even though the mechanism of revealing information is the same in both the small N and large N cases.
To formulate this cognitive bias clearly: we’re sense a change in odds more clearly when the magnitude of the change is large, versus when the odds only change slightly. Even though the mechanism by which the odds shift tends to be the same in both cases — new information that alters the probabilities is introduced in the same manner — we’re better able to interpret the effect of the information when it’s impact is large, and less capable of interpreting the information correctly when it’s less significant.
I’ll go even further and suggest that our ability to interpret the information is non-linearly dependent on the size of the set of doors. When there are 3 doors it’s confusing, when there are 5 doors the problem already starts to become tractable to our intuition, it’s more obvious that there’s benefit to switching. We get decreasing returns to clarity as the N increases. I already understand I should switch when there are 100 doors, and having 1 million doors doesn’t make it that much clearer. The confusion seems to be particularly strong when n is very small. This non-linearity suggests that perhaps in the very small n problem (i.e. when there are only 3 doors), there are actually differing intuitions about probabilities competing with each other and that’s what causes the short-circuit in our judgment. What these competing principles might be I haven’t reflected on in-depth.
As an observation, the ability to understand the probability more clearly when the number of doors is large also seems to have some relationship to our understanding of the law of large numbers: “The average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.” There’s an interesting paper on people’s actual intuitions around large sample sizes.
Our shortcomings in assimilating new information’s correctly when its impact is less pronounced is an interesting phenomenon that should have implications for how we design things to make them more obvious and intuitive. With computers, we’re able to run simulations that can visualize repeated trials of the same problem and their cumulative outcome, to see from the larger sample sizes more clearly what the probable outcomes of taking different actions are. Consider again Erdos convincing himself that the correct action in the 3 door example is switching by virtue of simulation.
Computers can also let us visualize a change in the input size of the problems quickly, to be able to rapidly see what the impacts of changing N is on the outcomes. This type of ability to scrub the parameters of the problem instance (in this case the size of N), and also to simulate running the problem many times, gives us a flexible model of the abstracted version of the problem, that should be able to help us get a better intuitive understanding of it across all instances. In my mind, this is an extension of the work a lot of people, like Bret Victor, have done on the direct visual manipulation of problem models and inputs for better understanding. Brett categorizes his research as Explorable Explanations — here are 3 great links to some of his relevant work:
http://worrydream.com/#!/ExplorableExplanations,
http://worrydream.com/#!/TenBrighterIdeas
http://worrydream.com/#!/ScientificCommunicationAsSequentialArt
What I’m proposing is that probability abounds with cognitive biases. Visual representations, directly manipulatable parametrized versions of problems, and simulation, can be particularly fruitfully applied when enabling explorable explanations of probability. An interesting research direction is identifying design principles that can work to counteract our cognitive biases, and help us assimilate deeper intuitive understandings of probability and other non-obvious aspects of the world.
While there’s something unsettling about computers directly telling us a truth, there is something more palatable about using computers to construct explorable models that correct our cognitive biases and fortify our understanding.
