Let’s revisit why we’re all gathered here today. We have been talking about how numbers can fail and obvious logic isn’t all it’s cracked up to be. We’ve talked about the average of averages, Simpson’s Paradox, and, most recently, a few quick questions that show us how we can be led down a logical path that is not always correct. Let’s look at the solutions for those last questions. (I’m not going to repeat them here, just go back to the previous post to play along.)

Okay, I admit this first one was pretty easy. But I’m willing to bet you’ve seen it happen. A 60 percent decrease followed by a 70 percent increase does not mean an overall increase. (Have you noticed how percentages play such a role in these deceptions?) The quick, easy way to prove this is to assume production was originally 100. A 60 percent decrease means production is now 40. A 70 percent increase on 40 is (approximately) 3, with a new production of 43 — not the greatest result in the world.

The second problem was how we could start with “1 equals 1” and wind up with “1 equals 2.” At first blush it seems that each algebraic step of this problem is legitimate. Yet we wind up with an illogical conclusion. The problem lies in the step where we divide both sides by a - b. If a = b, then a - b = 0. Division by zero cannot be defined and, accordingly, is not allowed. (And now you know just one reason why.) This is a perfect example of why you should always be sure you understand what underlies the steps that are being taken.

In the final problem, the throw most likely to occur is the first one — RGRRR. For those of you who got it right, you may be wondering what the big deal is. Well, studies have shown that the preference is for the second series — GRGRRR. All three sequences are heavily unbalanced toward the more unlikely red, and people explain that sequence 2 is somehow more “balanced,” or less unbalanced, and therefore more probable. It is a case where typical seems more likely. To show how people, in spite of being shown how things should really work, stick with their misconceptions, 65 percent of the people in the study still showed a strong attraction to sequence 2, even when it was explicitly pointed out that one can obtain sequence 1 from sequence 2 just by eliminating the first throw of the dice. (I’ll leave it up to someone else to crunch the numbers and prove the truth.)

Our minds lead us down the primrose path. Seventy is greater than 60, so production must be better. Algebra shows that we can divide both sides by the same number, so we don’t have to dig deeply to understand what that specific number is. And, a sequence contains a series of results we expect, so the entire series must be more likely to occur (even if the other series has fewer items in it). We jump to incorrect conclusions.

With that warning in mind, let’s go into some of the thornier issues we will discuss. We’ll start with “The Three Prisoner Dilemma.” Three prisoners face death. However the tyrant who has condemned them decides to spare one. The tyrant knows which he will spare, but decides, in his twisted way, to enjoy himself by not sharing the information with the prisoners. He does tell the guard. One of the condemned men, Prisoner C, speaks to the guard. “There is no doubt that one of the other two prisoner’s will be shot. Give me the name of one of the two who is certain to be shot.” The guard (as sadistic as the tyrant), tells him Prisoner A will not be shot. Prisoner C is happy. He feels that his odds of being spared have increased from a one in three chance to 50/50. His odds of survival have increased.

Is he correct?

Work on this a while, and let’s see some guesses (or logical conclusions) out there.

Ben(2/23/10 12:35 PM)Mike,

I tried to follow what you implied but couldn't get past your second paragraph because a 70% increase for 40 is 28 [not 2.8 or " . . . (approximately) 3 . . . " as you stated]. I promise that is no "harsh criticism" so please don't take it personally. I will read your article again when I have more time.