Mathematics Awareness Week 1996

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Probability and Intuition

by Peter Winkler

Probability and Intuition

Most of us don't use mathematics in everyday decision making---at least, not explicitly. We think of mathematics as a tool for making technical decisions in highly complex but fairly well-defined situations: how to build an airplane wing, how to distribute pension fund investments, how to route telephone calls on Mother's day.

But 99% of the decisions we make---including most of the important ones, as well---are made in a non-technical environment with the benefit of our human experience and intuition. Should you take an umbrella to work today? Should you let Fast Eddie borrow the car? Should you talk to your boss about you-know-what, or let it go? Should you go to that expensive, private college or the state school? Should you buy a lottery ticket? Life insurance? Cough medicine? Shoes? Caviar? Condoms? Should you ask that important person to marry you? Divorce you? Go to Paramus with you? Should you hire a research mathematician to help you, your family or your company make critical decisions?*

Actually, you already have mathematicians working for you. By studying decision-making in well-defined, quantitative situations, they help to guide people's intuition, keeping it from heading off in the wrong direction. Intuition is a valuable, indeed indispensable, commodity but it is not perfect and occasionally needs a helping hand.

Most of the decisions we make involve unknown factors, and thus involve (directly or indirectly) the science of probability. Unfortunately, it is in dealing with probabilities that intuition often falls short. Here are a few examples of places where MY intuition is crummy---see if yours is any better.


Despite many warnings to the contrary (e.g. "The best-laid plans..."), people persist in believing that if each event in a list is probable, then they will probably all occur. When I was a child a brewing company rented my home-town bowling alley and tried to film a commercial in which ten professional bowlers bowled strikes at the same time. These pros were good, but after a whole day of bowling the crew gave up in disgust.

Now, when you back blindly out of that driveway onto Elm Street, maybe it's 999 to 1 that you won't hit something. But if you do that every day...?

A bet that everything will happen the way you expect it to is called a "parlay" and you know, if you bet on horses, that it rarely works. Even if those bowlers each miss only one time in three, the probability of getting those ten simultaneous strikes on a given attempt is 2/3 to the tenth power, which is about 1/58. Does that mean if you try it 58 times you will succeed? No, you will be unlucky more than a third of the time.

A famous example of the unlikelihood of a parlay is the "birthday problem." The odds of two random people NOT having the same birthday are about 364 to 1, but put 23 people together and there are 253 pairs to consider; the probability of having no "coincidence" drops below 1/2.

Unequal Likelihood

People sometimes have a tendency to believe that similar-sounding events must be equally likely. Suppose you put three coins---a fair coin, a two-headed coin, and a two-tailed coin---into a sack and choose one blindly at random. You flip it and it comes up "heads". What is the probability that there is a head on the OTHER side of this coin?

Yes, it could be the fair coin or the two-headed coin, but they're not equally likely: because the fair coin COULD have come up tails, the two-headed coin is now twice as likely.

Your intuition is not totally off-base here, just slow. If you flipped the coin 10 times and got "heads" every time, you'd begin to suspect it was probably the two-headed coin. The point is that that inference exists already after one flip.

This problem is similar to the famous question of Monty Hall and the three doors. As a contestant on the TV show "Let's Make a Deal," you are faced with three doors, one of which conceals a valuable prize. You choose "A" but instead of opening it, the host Monty Hall, who knows where the prize is, opens (say) "B" to reveal a bag of peanuts. You are now given the opportunity to switch from "A" to "C". Should you?

Probable and Most Probable

Our last category concerns the tendency to confuse events which are "probable" with those which are merely the most likely of a large set of events. Here's an example: a coin is tossed 100 times. Is it likely that the result will be 50 heads and 50 tails?

Indeed, 50 is the most likely number of heads; but the probability of this outcome is only about 8%. So what? Well, among other things, you should certainly NOT conclude that a coin is biassed if this experiment does not yield exactly 50 heads and 50 tails. How many heads should make you suspicious? That is the subject of statistics, and depends both on how willing you are to impugn the coin unfairly, and how likely you think it was initially that the coin was biassed.

Of course, the winner of "most likely outcome" title is subject to change as more information comes in. Most people realize that flipping 4 heads in a row with a fair coin does NOT make it more likely that the next flip will be tails; it DOES make it more likely that you will end with 52 heads in your hundred flips, as opposed to 50.

Perhaps you have heard of the misguided gentleman who, out of fear of terrorists, carries a bomb with him on an airplane trip. His reasoning: after all, what are the odds that TWO people will carry bombs on a plane? Laugh, but many of us relax after one awful thing has happened, thinking that another is unlikely on the same day.

How Do We Learn?

How does mathematical knowledge reach the public, in order to help guide intuition? A big factor is games. In structured situations the odds can be computed, and people who need to calculate or at least memorize those odds can carry some of what they learn into everyday life.

We have already indicated that a person who likes to bet on horses might be more aware of the unlikelihood of parlays; sadly, he might well bet on them anyway. A backgammon player who accepts a double and plays on for a specific dice combination will soon learn her folly, however.

Interestingly, a bridge player is more likely to make the right decision on "Let's Make a Deal." Why? Because the "principle of restricted choice," critical in the game of bridge, says that a lie of the cards which HAD to have resulted in the play seen so far becomes more likely than one which only MIGHT have. On the game show, if the prize is behind door "C" he HAD to open "B", whereas if it's behind "A" he had a choice. So, the contestant should switch.

Of course, games will teach you nothing if you don't pay attention to the odds and learn to deal with them. A course in basic combinatorial probability might go a long way toward helping someone hone his or her intuition, and it can be a lot of fun as well.

Here's a test. Your teacher offers to bet you even money that if she rolls six dice, the number of different numbers which appear will be exactly four. Should you take it?

Well, if you've read this far, you may reason as follows: getting exactly four different numbers may be the single most likely outcome, but that doesn't mean it is probable. Sounds like a good bet to take.

But in fact the probability of getting exactly four different numbers on the six dice is better than 50%. Sometimes, there's no substitute for computation!

*Answer: yes.

Dr. Peter Winkler
Mathematical Sciences Research Center
AT&T Bell Laboratories
Murray Hill, NJ
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Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.