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Leonard Mlodinow: The Three Laws of Probability
In his book, The Drunkard's Walk, Leonard Mlodinow outlines the three key “laws” of probability.
The first law of probability is the most basic of all. But before we get to that, let's look at this question.
Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.
Which is more probable?
Linda is a bank teller.
Linda is a bank teller and is active in the feminist movement.
To Kahneman and Tversky's surprise, 87 percent of the subjects in the study believed that the probability of Linda being a bank teller and active in the feminist movement was a higher probability than the probability that Linda is a bank teller.
1. The probability that two events will both occur can never be greater than the probability that each will occur individually.
This is the conjunction fallacy.
Why not? Simple arithmetic: the chances that event A will occur = the chances that events A and B will occur + the chance that event A will occur and event B will not occur.
The interesting thing that Kahneman and Tversky discovered was that we don't tend to make this mistake unless we know something about the subject.
“For example,” Mlodinow muses, “suppose Kahneman and Tversky had asked which of these statements seems most probable:”
Linda owns an International House of Pancakes franchise.
Linda had a sex-change operation and is now known as Larry.
Linda had a sex-change operation, is now known as Larry, and owns an International House of Pancakes franchise.
In this case it's unlikely you would choose the last option.
If the details we are given ﬁt our mental picture of something, then the more details in a scenario, the more real it seems and hence the more probable we consider it to be—even though any act of adding less-than-certain details to a conjecture makes the conjecture less probable.
Or as Kahneman and Tversky put it, “A good story is often less probable than a less satisfactory… [explanation].”
2. 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.
Suppose a married person has on average roughly a 1 in 50 chance of getting divorced each year. On the other hand, a police officer has about a 1 in 5,000 chance each year of being killed on the job. What are the chances that a married police officer will be divorced and killed in the same year? According to the above principle, if those events were independent, the chances would be roughly 1⁄50 × 1⁄5,000, which equals 1⁄250,000. Of course the events are not independent; they are linked: once you die, darn it, you can no longer get divorced. And so the chance of that much bad luck is actually a little less than 1 in 250,000.
Why multiply rather than add? Suppose you make a pack of trading cards out of the pictures of those 100 guys you’ve met so far through your Internet dating service, those men who in their Web site photos often look like Tom Cruise but in person more often resemble Danny DeVito. Suppose also that on the back of each card you list certain data about the men, such as honest (yes or no) and attractive (yes or no). Finally, suppose that 1 in 10 of the prospective soul mates rates a yes in each case. How many in your pack of 100 will pass the test on both counts? Let’s take honest as the first trait (we could equally well have taken attractive). Since 1 in 10 cards lists a yes under honest, 10 of the 100 cards will qualify. Of those 10, how many are attractive? Again, 1 in 10, so now you are left with 1 card. The first 1 in 10 cuts the possibilities down by 1⁄10, and so does the next 1 in 10, making the result 1 in 100. That’s why you multiply. And if you have more requirements than just honest and attractive, you have to keep multiplying, so . . . well, good luck.
And there are situations where probabilities should be added. That's the next law.
“These occur when we want to know the chances of either one event or another occurring, as opposed to the earlier situation, in which we wanted to know the chance of one event and another event happening.”
3. 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 percent).
When you want to know the chances that two independent events, A and B, will both occur, you multiply; if you want to know the chances that either of two mutually exclusive events, A or B, will occur, you add. Back to our airline: when should the gate attendant add the probabilities instead of multiplying them? Suppose she wants to know the chances that either both passengers or neither passenger will show up. In this case she should add the individual probabilities, which according to what we calculated above, would come to 55 percent.
These three simple laws form the basis of probability. “Properly applied,” Mlodinow writes, “they can give us much insight into the workings of nature and the everyday world.” We use them all the time, we just don't use them properly.