You Don’t Understand Probability

Let’s say you are considering putting money into a bet of $1,000. It’s a coin toss. Nobody is cheating. You know it’s absolutely fair. You know you will have either zero or $2,000 in your pocket by tomorrow night, depending on whether the coin gets tails or heads, each with a 50% probability.

The expected outcome of the bet, i.e., the probabilities of each payoff multiplied by the dollar values at stake is $1,000 (i.e., 50% x $0 + 50% x $2,000). Can you visualise owning $1,000?

Chances are you can imagine only one state at a given time, either 0 or $2,000. Only one of the states would dominate the picture—the fear of ending with nothing or the excitement of an extra $1,000. You are likelihood to bet depends upon which state dominates.

Let’s take one more example. Say you’ve been going to the doctor due to some complications. You’ve run some tests. The results have come out today. Your doctor says in a grave tone, “Umm…I’m afraid it’s cancer!” The message hits you! It hits your body along with your mind. You figured something was wrong even before he uttered it, but cancer, really? You think. Nobody that you know of, your colleagues, family, friends have got cancer. Why you?

“But it’s not as bad as it sounds,” your doctor continues. “It’s a rare form of cancer with a 68% survival rate in 3 years. Go home and get you stuff in order. We begin treatment tomorrow.”

68% survival rate. That means 68 people of 100 make it. And it takes around 3 years without clinical manifestations of the disease for the patient to be pronounced cured. That is not so bad, you think. You now feel in your guts quite certain that you are going to make it.

In your mind a 32% chance of death meant the image of you dead. A 68% chance of survival is bound to put you in a cheerful mood.

It’s very unlikely that you stop to wonder about the mathematical difference between a 32% chance of death and a 68% chance of survival. Clearly, there is none, but you are not made for mathematics. In your mind a 32% chance of death means the image of you dead. A 68% chance of survival is bound to make your cheerful.

You might already start planning how you are going to reset your whole life once you are cured. At no point during his ordeal you do think of yourself as 68% alive and 32% dead. Your mind doesn’t understand probability. Your brain can properly handle one and only one state at once. Even if the odds are 99 to 1.

In probabilistic situations, decision making is not purely rational—because you decide mostly based on what you can picture in your head.

When confused between two vacation spots, you picture each of them individually (giving each a 100% probability) in your head, and whichever evokes better emotions gets picked.

If you are waiting for your entrance results tomorrow, and there’s (roughly) a 75% chance that you are going to make it and a 25% chance that you are not, it’s really hard to be 75% happy and 25% sad at the same time. You are either tensed or sipping a chilled beer. Not only that, your decision also depends upon how the information is presented.

In 1999, Sally Clark, an English solicitor went on trial for the murder of her two infant sons. She claimed they both succumbed to sudden infant death syndrome (SIDS). When consulted, Sir Roy Meadow, a renowned paediatrician, argued that the odds of SIDS claiming two children from a family were 1 in 75 million. The jury convicted Clark to life in prison.

The problem lies in how the data had been presented. Meadow presented his evidence in the natural frequency format (for example, 1 in 10 people), rather than in terms of a percentage (10 percent of the population). 1-in-10 is more intuitive and impactful. The probability format is more abstract and less intuitive. You don’t understand probability.

However, after four years in prison, it was proven that there was “no statistical basis” for Meadow’s stated figure. It was just an expression to denote it’s unlikelihood. Clark’s conviction was overturned on appeal, and the case has become a canonical example of the consequences of flawed statistical reasoning.

The inability to process probability also explains why you can’t grasp odds very well. When you consider the chances of, say, coming down with a specific illness as 1 million to 1, it seems unlikely. You interpret ‘unlikely’ as ‘impossible’. What you don’t realise is that in a city with a million people, that could easily be you.

On the flip-side, those odds also mean it would be irrational to completely change your lifestyle or habits just to avoid contracting that illness, since those odds are actually very less likely.

To elaborate our inherent inability to understand probability, Andrew Gelman, a professor at Columbia University conducts a simple lab experiment.

“As a way of giving students an intuition into randomness, we divide the class and then I leave the room. One half of the class has to create a sequence of 100 coin flips, and then they’re told to write it on the blackboard as a sequence of zeros and ones. The other half of the class was instructed to create a fake sequence of 100 zeros and ones that’s supposed to look like coin flips.”

“I return and what I see is two blackboards, each with sequence of 100 zeros and ones, and I can immediately tell which is which.”

“A real sequence of coin flips is likely to have a long run of heads or tails. You’re more likely than not to see a run of seven straight heads or seven straight tails. The fake sequence, they tend to alternate too much between heads and tails, so I can immediately see the fake one looks too random and the real one looks like something strange went on.”

Whenever you get seven straight heads or tails, you might wonder, what are the odds of that?

Similarly, have you ever gone to the phone to call a friend only to have your friend ring you first? What are the odds of that? Not high, to be sure. But given enough opportunities, enough phone calls—this is bound to happen once in a while.

A similar misconception of probability arises from the random encounters you may have with relatives or friends in highly unexpected places. “It’s a small world!” you often utter with surprise. But these are not improbable occurrences—the world is much larger than you think.

It is just that you are not truly testing for the odds of having an encounter with one specific person, in a specific location at a specific time. Rather, you are simply testing for any encounter, with any person you have ever met in the past, and in any place you will visit during the period concerned. The chances of the latter is considerably higher.

Now, does all these mean that you will perpetually remain stuck when it comes to risk and probability? Possibly not. Language is very important when it comes to dealing with probabilities.

Let’s describe the probabilities associated with a test for an illness in an example. Your doctor says, “One percent of people tested have the disease, and the test is 90 percent accurate, with a nine percent false positive rate.”

With all that information, what do you deduce about the likelihood of you having the disease if you have tested positive? Think about it for a moment. Now, let’s rephrase the statistics: “If we ignore the negative tests, nine times out of ten, a positive test for the disease is a false positive.”

Put that way, it’s easy to understand that even if you’ve got a positive result in the test, there’s only a ten percent chance that you have the disease.

Thus, the use of language makes all the difference. This is both good and bad. There’s not a lot you can do if someone decides to intentionally target your brain’s weak spots. This inability to process probability gives way to get fooled by conmen, profiteers, lawyers, and even by your own self. After all, you don’t understand probability.