Expected value
📐FORMULA
$$E[X] = \sum_{i} p_i,x_i$$
- xᵢ: each possible outcome, as a number
- pᵢ: the probability of that outcome; the pᵢ must add up to 1
- E[X]: the probability-weighted average of the outcomes
It is what I would get on average if I could make the same bet many times. For a single bet it may be a value that can never actually occur: the expected value of one die is 3.5.
It came in at the end of 🎓 Lecture 4 - Conditional probability, on the back of the probability trees: multiply along the branches, add at the leaves of the tree.
What it leaves out
- The spread. Two options with the same expected value can differ wildly in how bad the bad case is.
- Whether I get to repeat it. An average over a thousand tries is cold comfort for a decision made once.
- How the outcomes feel. Losses weigh more than gains of the same size, which the formula politely ignores.
So in a 🧭 One-page decision memo it goes under options as one line per option, with the worst case written next to it. It is an input to the recommendation, never the recommendation.
Worked example
The exercise I used in the pricing workshop, with round invented numbers so nobody argues about the data. A small studio can quote a job two ways:
| Quote | Chance the client says yes | Expected value |
|---|---|---|
| 6,000 | 0.50 | 0.50 × 6,000 = 3,000 |
| 4,500 | 0.80 | 0.80 × 4,500 = 3,600 |
On expected value the lower quote wins by 600. Then the room found what the sum had not been told:
- At 4,500 the studio is busy 80 percent of the time and has no slack for the better job that arrives in week three.
- The probabilities were guesses. Move the first one from 0.50 to 0.60 and the two options tie at 3,600.
- The lower price becomes the reference point for the next quote.
The useful output was not 3,600. It was the question how sure are we about that 0.50, which took the rest of the hour.