CalculateQuestion 1 · Apply
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By the end you can calculate from a probability distribution table, explain what the number means, and use it to decide whether a game is fair.
The main symbols used in this lesson:
| Symbol | Meaning |
|---|---|
| the random variable (for example, the score on a die) | |
| one particular value that can take | |
| the probability that takes the value | |
| the expectation (expected value) of | |
| the mean of ; the same number as ("mu", Greek m) | |
| "add up", over every possible value |
The expectation is the average value of in the long run. It is also called the mean of , written .
You already know how to average a list of numbers: add them and divide by how many there are. In a distribution, though, some values happen more often than others. A value with probability turns up four times as often as one with probability , so the values cannot count equally.
Instead, each value is weighted by its probability, that is, by how often it occurs in the long run:
In words: multiply each value by its probability, then add.
Notice that need not be a possible value of . A single die has expected score , though no roll ever shows .
Imagine rolling a fair die times. Each score turns up about times.
Dividing by is the same as multiplying each score by , its probability. So "value times probability, then add" is just the ordinary average of a very long run of trials.
So is where the average would settle if you repeated the experiment endlessly. For a single trial it is a prediction of the centre, not of the outcome.
Worked example: Example 1: expectation from a table
The discrete random variable has the distribution below. Find .