Expected Value

Not sure if this is the best forum for feedback, so please direct me elsewhere and happy to delete my comment if not. 

A few suggestions on the explanation of EV. While the examples are clear, I found the definition of expected value confusing. 

It is written as "expected value = likelihood of option x value of option", and "The expected value is the probability multiplied by the value of each outcome".

I read this as: $E\left[X\right]=xP\left(x\right)$, which doesn't capture the need to sum across outcomes.

Pitched at the same level of technicality, I think a clearer definition is: "The expected value of an uncertain decision is the sum across all outcomes of the value of each outcome multiplied by its probability." 

Or some other wording that captures that this is a weighted average. This properly implies the necessary summation across outcomes: $E\left[X\right]=\sum xP\left(x\right)$.

It might also be worth:

• Giving detail to the calculation of the expected value of "If you have a 50% chance of winning a coin flip for a $1 coin, the expected value is 50 cents". Because it has a 0-weighted outcome, it isn't obvious that you need to summate across outcomes here, which is confusing if you don't see the application of the formula: $EV=0.5\ast 0+0.5\ast 1=0.5$.
• Giving EV an interpretation, e.g. "This has the interpretation of a long-run average, or what we would expect if we were able to repeat our plan many times." This would help to ground the motivation for using EV in the "what should you do" section.
• Noting that this only holds in the discrete case

This one now requires you to sign up, which is a bit unwieldy. I went for the 80k article instead, but let me know if there's an easier way!

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