What is Expected Value?

Welcome back friends!

Today we are going to talk about another probability and statistics concept called Expected Value.

Expected value is what you think it is: the return you can expect given the data or knowledge you already have.

Investors use expected value all the time. They make decisions to buy/sell a stock based on what the expected value is for that stock.

Without thinking about it, we intuitively use expected value when we make everyday decisions. We factor in the benefits and risks of a decision and if it has a positive expected value, we usually are in favor of that decision, else we are against it.

For example, when we take on a new project or (a blog post in this case), we view the expected value in terms of personal development and other career benefits as higher than the cost in terms of time and/or sanity.

Likewise, anyone who reads a lot knows that most books they choose will have minimal impact on them, while a few books will change their lives and be of tremendous value.

Looking at the required time and money as a cost, reading books has a positive expected value.

Back to math

Expected value informs us what we think “the long-term” average will be after adding many more trials or records.

For example, if we flip a quarter 10 times, it’s probably not going to be 50/50. If you flip it 100 times, it’s still probably not going to be 50/50, but closer. But the more and more you flip the coin, the closer you will get to the 50/50 expected value of coin flipping.

Another great example of this is rolling a die. As we increase the sample size, we see that the probability of rolling any value approaches 1/6.

Notation

I had to throw in some notation here for completeness. For discrete data (non-continuous), it’s pretty easy. It’s just the weighted average of the possible values and their respective probabilities.

μ = E(X) = ∑[x·P(x)]

where

μ = mean

E(X) = expected value

x = an outcome

P(x) = probability of that outcome

A company makes electronic gadgets. One out of every 50 gadgets is faulty, but the company doesn’t know which ones are faulty until a buyer complains. Suppose the company makes a $3 profit on the sale of any working gadget, but suffers a loss of $80 for every faulty gadget because they have to repair the unit.

E(X) = 49/50 • 3 + 1/50 • (-80)

= 147/50 – 80/50

= 67/50

= 1.34

The expected value is $1.34 on every gadget made, and since its positive, we can expect the company to make a profit.

Finding the expected value of a continuous variable – like one from a normal distribution – is a little more complex involving calculus. We’ll get into that later.

That’s all for now!

-Paul

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