These three laws, simple as they are, form much of the basis of probability theory. Properly applied, they can give us much insight into the workings of nature and the everyday world.
– Leonard Mlodinow
That quote is from Leonard Mlodinow’s book, The Drunkard’s Walk: How Randomness Rules Our Lives. The book contains examples as varied as politics, wine ratings, and school grades to show how a misunderstanding of probability causes people to misinterpret random events. Mlodinow’s three laws of probability are as follows:
- The probability that two events will both occur can never be greater than the probability that each will occur individually.
- If two possible events, A and B, are independent, then the probability that both A and B will occur is equal to the product of their individual probabilities.
- If an event can have a number of different and distinct possible outcomes, A, B, C, and so on, then the probability that either A or B will occur is equal to the sum of the individual probabilities of A and B, and the sum of the probabilities of all the possible outcomes (A, B, C, and so on) is 1 (that is, 100%).
When we don’t understand probability, we fall prey to the conjunction fallacy. As I previously wrote,
we might hear separate rumors that corporate budgets will be cut soon and that the senior executive for our department is considering leaving the company. We judge each of these events on their own as unlikely – perhaps a 33% chance of budget cuts (the company is doing well) and a 25% chance of the executive leaving (she’s been here for 10+ years). But when we hear both rumors, our intuition that both events will happen is quite high – perhaps 50% or more. As a result, we spend more time than we should worrying about the funding for our project and maybe even update our résumé.
Assuming the executive is not leaving because of the budget cuts (i.e. the events are independent), the probability of both happening is 0.33*0.25 or only about 8% – not likely at all. Even if the events are related, by law 1 the probability of both happening cannot be more than 33%.
The Drunkard’s Walk provides another example based on empty airline seats which I’ve modified to strengthen the point. Imagine an airline has only one seat left on a flight and two passengers have yet to show up (they’ve overbooked the flight). From experience, the airline believes there is a 75% chance a passenger who books a seat shows up in time. Mathematically, overbooking makes sense if your goal is to fill the plane: the chance that neither will show up and the plane will fly with an empty seat is very low: 0.25 * 0.25 is 6%. On the other hand, it’s risky from a customer experience point of view: There is a 0.75 * 0.75 = 56% chance both show up and they must deal with an unhappy customer. From law 3, the probability that it all works out perfectly and one (and only one) person shows up is less than 38% (1 – 0.56 – 0.06). These are not great odds and yet airlines do this all the time.
Of course, the above assumes the passengers are independent. If they are traveling together, the situation is even worse. The chance both people will show up is 75% and that neither shows up is 25%. There’s literally no chance that exactly one person shows – the situation the airline is counting on. A combination of ignoring customer experience and not understanding probability might explain why we’ve had so many unfortunate airline incidents recently.
Yes, I realize this post had more math than you might be used to in my writing. The same is true with the book. But that’s sort of the point: we all need a little better understanding of probability if we want to make sense of our surroundings.
Or as Mlodinow writes, “probability is the very guide of life.”