"Critical Thinking"... A Simple Example

I’ve referenced “critical thinking” lately in both blogs, so figured I’d end the week with a simple example from Daniel Levitin’s timely book (“A Field Guide To Lies”) that I’ve also been touting. Many readers will already be familiar with this probability case in one form or another, but still instructive, especially for those unfamiliar. Levitin sets it up as a street vendor playing a betting game with passersby, more-or-less like this:

The vendor has 3 cards he shows you, one that is white on both sides, one that is red on both sides, and one with a red side and a white side. (And you can play the game as many times as you like.)

Again, the cards are:

Red              White            Red

———        ———-        -———

Red              White            White

He puts the cards in a hat, pulls one out and places it on the table showing a red side; then says, “I’ll bet you $5 the other side is red.” Should you take the bet? The passerby may think, “well, sure, I have a 50/50 chance, since 1 card (the white/white) is eliminated, I’m looking at either the red/white or the red/red card. So there’s a 50% chance the other side is white, and even if I lose I can play the game again, and maybe get my money back.”

Seems common-sensical, but it is critically wrong.

There are actually THREE ways a red side can be showing: one way from the Red/White card and two different ways from the Red/Red card. Thus, there is a 2/3 (not 1/2) chance the other side is red — and of course the same probabilities hold if the shown side is white and the vendor says he’ll bet the other side is white. The vendor (no surprise) has a distinct advantage.

Anyway, it’s a simple little gambit that will fool a lot of people, and yet once explained, with just a bit more critical thought, the situation is clearcut.

Probability is a particularly rich area for such ‘critical thinking’ examples, that sometimes even fool professional mathematicians. Is it any wonder that the rest of us are so easily blind-sided.

Of course confusion with numbers seems a far cry from confusion with politicians' rhetoric, but in both cases there is the need for care and clarity of thought in evaluating.