Quiz time!

Quiz time! Don't worry, there are no wrong answers. But please think about and answer each question in turn, before moving on to the next question.

Question #1: Which of the following would you prefer to have?

(A) A 100% chance of winning $1 million

— or —

(B) An 89% chance of winning $1 million, a 10% chance of winning $2 million, and a 1% chance of winning nothing?

Question #2: Which of the following would you prefer to have?

(A) An 11% chance of winning $1 million and an 89% chance of winning nothing

— or —

(B) A 10% chance of winning $2 million and a 90% chance of winning nothing?

Question #3: For this question, imagine there's a box in front of you. You have no idea what's in the box: It could be something good, such as a billion dollars; or it could be something bad, such as a poisonous spider; or it could be something neutral, such as a used pencil. With that in mind, which of the following would you prefer to have?

(A) An 89% chance of winning whatever is in the box and an 11% chance of winning $1 million

— or —

(B) An 89% chance of winning whatever is in the box, a 10% chance of winning $2 million, and a 1% chance of winning nothing?

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If by now you suspect these are trick questions, you're right. While there's no wrong way to answer the questions, all the questions taken together have only two rational sets of answers: all A or all B. Any mixing of A and B answers leads to a contradiction. Here's why.

The first question is entirely based on preference: would you rather have the sure thing or take a small risk to go for a bigger gain? There's no correct answer.

Question #2 phrases the same question differently by removing an 89% chance of winning $1 million from each choice. However, for many people who answerA to Question #1, the same choice seems too prudent for Question #2. Why increase your chance of winning by a mere percentage point at the cost of giving up half the winnings?

Question #3 shows the similarity between the two previous questions by replacing the missing 89% chance with a mystery box. The box shouldn't affect your answer because both the A and B answers give an identical 89% chance to win the box. So you should decide which answer to pick based on the remaining odds: an 11% chance to win $1 million versus a 10% chance to win $2 million—the same choice in Question #2.

However, the contradiction is that the same reasoning works for Question #1, too. To see that, imagine that Question #1 were phrased as follows:

Question #1-B: Which of the following would you prefer to have?

(A) An 89% chance of winning 1$ million and an 11% chance of winning $1 million

— or —

(B) An 89% chance of winning $1 million, a 10% chance of winning $2 million, and a 1% chance of winning nothing?

Question #1-B is the same as Question #1, and it's also the same as Question #3 but with the mystery box replaced with $1 million. Therefore, based on the similarity we already established between Question #2 and Question #3, all three questions are asking the same thing with the one difference of what's being offered at an 89% chance: $1 million, nothing, or a mystery box. And the 89% chance shouldn't affect your choice in any of your answers, so therefore you should choose the same answer for all three questions.

If you live on Planet Rational.

These questions make up what's called the Allais Paradox. I've lifted it from another William Poundstone book I've started reading—this one called Priceless: The Myth of Fair Value (and How to Take Advantage of It).