Expected Return: It's Not What You Expect

Instead of having "answers" on a math test, they should just call them "impressions," and if you got a different "impression," so what, can't we all be brothers? - Jack Handey

I recently had the pleasure of watching an interesting discussion unfold on the Vanguard Diehard message board. The discussion involved a hypothetical investment which returned either 30% or -10% with equal probability. Several parties debated the expected return of this investment and each participant gave a convincing argument. This is math, however, and there is only one correct answer. Read on to find the definitive answer to this conundrum, if you dare.

Many thanks go to William J. Bernstein and the other participants in the original conversation. This article is based on their insightful commentary. The original thread can be viewed here

Probability Is A Harsh Mistress

Before we get started let's get one thing straight: Probability is hard. Probability is incredibly difficult for humans to understand. In many cases it is confusing and counter-intuitive. Occasionally probability is natural and straight-forward. But when probability makes sense, the occasion is so remarkable that one immediately assumes a mistake must have been made. Like an action hero walking into a quiet room, there must be some detailed that is being overlooked. It's quiet in here. Too quiet.

If you think probability is easy then you are either much smarter than I am or you are deluding yourself, and the probability is very high that it is the latter option. But don't feel bad. If you think probability is difficult then you are in good company: The rest of the human race. To demonstrate the confusing nature of probability let's take a short test. The answers will be shown on the next page. No cheating.

1. Which is most likely to occur? A family that has 6 boys for children, or a family that has 3 boys and 3 girls? (Lets assume boys and girls are equally likely for each birth. In reality boys are born more often than girls with a ratio of about 1.06:1, and you are more likely to have multiple children of the same gender.)
   1. The family with 6 boys
   2. The family with 3 boys and 3 girls
   3. They are equally likely to occur


2. Which is most likely to occur? A family that has 6 boys for children, or a family that has 3 boys and 3 girls ordered by age from youngest to oldest as girl, boy, boy, girl, boy, girl.
   1. The family with 6 boys
   2. The family with 3 boys and 3 girls
   3. They are equally likely to occur


3. A fair coin is tossed 4 times. Which outcome is most likely?
   1. First toss is heads, second is heads, third is heads, fourth is tails.
   2. First toss is tails, second is tails, third is heads, fourth is tails.
   3. First toss is heads, second is tails, third is heads, fourth is tails.
   4. First toss is tails, second is tails, third is tails, fourth is tails.

Continue reading page 2 for the answers.