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Intuition Led Astray

3 March 2002

The American Heritage Dictionary defines intuition as "the act or faculty of knowing without the use of rational processes; immediate cognition." Webster's Unabridged Dictionary has a similar definition: "the immediate knowing or learning of something without the conscious use of reasoning; instantaneous apprehension." Of the two I like the Webster definition because of the last two words. When applying intuition to a problem, one should always maintain a healthy apprehension. In this article I should like to illustrate why I feel this is the case.

Let me begin by saying that intuition is a wonderful facility that we humans have. I believe that it helps keep us safe and steers us away from danger. It often keeps us from making foolish choices in our daily economic lives. Some feel it helps them detect lies. As a professional mathematician I use my intuition to make conjectures about what might or might not be a theorem in a particular field of study. And so on. Nevertheless, the application of intuition to many problems is fraught with pitfalls. Let's take a closer look.

To begin with I think the above dictionary definitions, while certainly describing an abstraction which I would call perfect intuition, do not accurately reflect the practical mechanisms of intuitively reached conclusions. I believe that actual intuition is a combination of human feelings combined with logic and data. This mixture of reason and emotion can be, I believe, the genesis of intuition gone astray. Here is a short list of common intuitive procedures that are faulty:

  • generalizing from a few examples to a general conclusion with too few examples
  • arguing to a general conclusion using analogies when the analogy is poor
  • using sound logic with incomplete hypotheses (missing data)
  • having complete data but using faulty reasoning
  • addressing the wrong question

Let's look at some examples of these.

I mentioned above that I use intuition in the pursuit of mathematics. How? Well, one way is by looking at lots of examples. Years ago I was trying to prove a particular conjecture in a branch of abstract algebra known as Lattice Theory. I had looked at many, many examples and in each one the conjecture held. Moreover, by looking at these examples I began to get a "feel" for the problem. I could sense why it had to be true. As the saying goes, "I could feel it in my bones." Yet it defied proof. I struggled off and on with it for two years. Then one day one of my colleagues who had been working on the same conjecture walked into my office and said "Guess what, here is a counterexample!" Crushed! But there is a bright spot to the story, namely, in those two years I never at any time did I think I had proved it. But my intuition was in error. Although I had looked at lots of examples, I hadn't looked at enough.

From my experience with students over the years I get the feeling that some people think mathematics was found under a rock somewhere. No indeed. It is the result of human beings using their intuition to guess at reasonable conjectures. Every theorem and definition in mathematics is the consequence of some human's intuition at some point in history. How do we know when they are right? That is what mathematics is for. It breaks down proofs into pieces small enough that we can get our heads around each step and logically conclude some complicated, non obvious fact that was the result of some persons intuition. It "keeps us honest" so to speak.

In the world of gambling, intuition can easily lead us astray. In October of 1999, on this web site, I wrote an article entitled Double Bonus - Break Up Those Pairs? The crux of the article was a question asked me by a Binion's Blackjack dealer about the Video Poker Game of 10/7 Double Bonus. He pointed out that when you are dealt two pair, one high and one low, that you might just as well throw away the low pair because the Double Bonus pay schedule pays 1 for 1 for both a high pair and two pair. Notice that the logic here is correct; throwing away the lower pair will not degrade the initial hand. The problem is that not all of the salient facts are included in the assumptions. Because the Full House pays 10 for 1 in this game, it turns out that tossing the low pair is not the correct play and I demonstrated that in the article.

Last month I discussed the role of Aces in 9/6 Jacks or Better Video Poker. Given a hand with three high cards, one of them an Ace, should one hold the Ace? Well, is the Ace the highest card in this game? The short answer is: yes, it is. Well then, just as in regular Draw Poker, it would seem reasonable, by analogy, that one should at least hold the Ace. As I showed you last month, this is wrong. The problem? The Ace is the highest card in 9/6 Jacks, but it only manifests its rank in one instance: the Royal has a higher payout than any other Straight Flush. Nowhere in the rest of the pay table does the ranking of the Ace matter one iota. A pair of Jacks is as good as a pair of Aces, three Twos is as good as Three Aces, a 5-high Straight is as good as an Ace-high Straight, and so on. Only by looking at the mathematics can we discern the correct answer. The analogy with regular Poker is a poor one.

Sometimes our intuition is fooled because we don't recognize the nature of the problem. In September of 1999 I wrote an article about sucker bets entitled An Ear Full of Cider and presented one of the all time great ones called The Birthday Problem. In a nutshell, the wager is to bet the dupe that in a roomful of, say, 30 people that at least two people will have the same birthday; the wager is for even money. Sounds like a good bet, doesn't it? After all, there are 365 days in a year and with a population of less than 10% of that number hitting a particular birthday twice seems very unlikely. Well, don't take that bet; as I show in the aforementioned article, the proposer has in excess of a 41% edge. How can that be? The problem is with the words "particular birthday." The bet just says that two people have to have the same birthday but does not specify the date and, as my article shows, that makes all of the difference in the world. One should always address the right question.

In the November 28, 1998 issue of Parade magazine, a letter was sent to Marilyn vos Savant, writer of the Ask Marilyn column, arguing that the casino game Chuck-a-Luck appeared favorable to the player. I addressed this letter in a column in May of 1999, Cardano's Gaff Lives On; you can read a transcript of the letter there or look up the original in Parade. As I demonstrated in my article, the writer's error is a common one in handling probabilities, namely, adding probabilities over non-disjoint events. The writer had all of the rules right, but his logic was flawed in that he tried to use a theorem from probability theory in a situation where the hypothesis didn't hold. If you have time, look it up; it's interesting reading.

If I had to pick the most common error in evaluating wagers it would have to be that of evaluating chances rather than expected returns. People often ask me questions like "What are the chances of ...?" or "What are the odds of ...?" or "What is the probability of ...?" rather than the correct question, which is "What is the expected return of ...?" They are asking the wrong question. Let me give you a dramatic illustration of this.

Suppose that we have two sacks that look identical. One sack contains nine $1 bills and one $10 bill and the other contains two $1 bills and one $100 bill. One of the sacks is randomly selected (probability 1/2) and two bills are drawn. If either the $10 bill or the $100 bill is drawn, the bill is returned to its sack and the sack drawing is repeated. Whenever two $1 bills are drawn from the selected sack, the player gets to pick and keep a third bill from either the sack from which the two $1 bills were drawn or from the other sack. Which sack should the player choose?

Here is a seemingly reasonable argument for always choosing the other sack. Since it is more likely that one would pick two $1 bills from the sack containing nine of them, that is the most frequently chosen sack when the game progresses to choosing the third bill. Thus picking the other sack will usually give us a one in three chance of getting $100 rather than sticking with the same sack and getting a one in eight chance of getting $10. Sound good? Let's take a closer look.

Recall from earlier articles (see, for example, the aforementioned An Ear Full of Cider) that the conditional probability of an event A given and event B, written P(A|B) is given by the formula

P(A|B) = P(A and B)/P(B) (1)

Using a bit of simple algebra, we can rewrite this as

P(A and B) = P(A|B)P(B) (2)

Interchanging the roles of A and B in (2) we obtain

P(B and A) = P(B|A)P(A) (3)

Since the event A and B is the same as the event B and A the numbers on the left hand sides of (2) and (3) are equal; hence the right hand sides are equal as well. Thus

P(A|B)P(B) = P(B|A)P(A) (4)

which we can rewrite as

P(A|B) = [P(B|A)P(A)]/P(B) (5)

This formula, though easily derived as you just saw, is very important in probability theory and is called Baye's Theorem. It is just what we need to analyze our problem.

Let us designate by sack #1 the sack containing the nine $1 bills and the one $10 bill, sack #2 is the other one. This done let me define the following three events:

A = sack #1 is chosen
A' = sack #2 is chosen
B = two $1 bills are selected from chosen sack

Right away we see that P(A) = P(A') = 1/2. Also if sack #1 is chosen, then there are 9 x 8 ways to choose two $1 bills in order and 10 x 9 ways to choose any two bills in order so that

P(B|A) = (9 x 8)/(10 x 9) = 8/10 = 4/5 (6)


P(B|A') = 1/3 (7)

Since the event B arises from either A or A' occurring, the probability of B occurring is the probability of B and A occurring plus the probability of B and A' occurring. From this observation and (3) we have

P(B) = P(B|A)P(A) + P(B|A')P(A') (8)

From (6) and (7), therefore, (8) gives us

P(B) = 4/5 x 1/2 + 1/3 x 1/2 = 2/5 + 1/6 = 17/30 (9)

We now have all of the pieces necessary to use Baye's Theorem (5) to calculate P(A|B) and P(A'|B). Here goes.

P(A|B) = [4/5 x 1/2]/(17/30) = 2//5 x 30/17 = 12/17 (10)
P(A'|B) = [1/3 x 1/2]/(17/30) = 1/6 x 30/17 = 5/17 (11)

So whenever two $1 bills are chosen, the probability that they were chosen from sack #1 is 12/17 and the probability that they were chosen from sack #2 is 5/17. This corresponds to our intuitive feeling on the matter, but importantly, it quantifies it.

Now suppose that our strategy is to always pick the other sack. If sack #1 was picked then our expected return from picking the other sack would be

exp (full sack #2) = 2/3 x 1 + 1/3 x 100 = 102/3 (12)

On the other hand if sack #2 had been picked then our expected return by choosing the other sack would be

exp(full sack #1) = 9/10 x 1 + 1/10 x 10 = 19/10 (13)

Weighting these by their frequency of occurrence we obtain the overall expected return by always picking the other sack:

exp(choose other) = 12/17 x 102/3 + 5/17 x 19/10 = 24.56 (14)

On the other hand suppose that we always pick the same sack. In this case if sack #1 is chosen the expected return choosing the same sack will be

exp(depleted sack #1) = 7/8 x 1 + 1/8 x 10 = 17/8 (15)

If sack #2 was chosen we would have

exp(depleted sack #2) = 100 (16)

since there are no $1 bills left. Weighting these by their frequencies we obtain

exp(choose same) = 12/17 x 17/8 + 5/17 x 100 = 30.88 (17)

This strategy is over 25% better than choosing the other sack! Contrary to our intuition you should always choose the same sack.

This is a dandy example. Our intuitive observations were correct as far as they went but we didn't ask the right question, namely, which strategy produces the highest expected return. I hope this makes my point. By the way, here is an interesting question. How much should you charge someone to play this game so as to insure a long run profit? The answer should be clear to you.

I should like to close this article by quoting a renowned mathematician by the name of George Polya. Professor Polya devoted a good bit of his career to addressing the issue of mathematical creativity; just how does mathematical intuition work? Professor Polya's remark was this: "No idea is really bad unless we fail to be critical of it. The bad thing is to have no idea at all." Good words by which to live. See you next month.

Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers