Probability: outcomes & distributions

Hook

If you had to describe how likely each outcome of something uncertain is, what rule do those likelihoods have to follow?

Intuition

Drag any bar. Watch what happens to the others — the whole picture always fills exactly one bar's worth of height, no more, no less.

Formalize

That's a probability distribution: a number for every outcome, following two rules —

P(x_i) \geq 0, \qquad \sum_i P(x_i) = 1

$P(x_i)$ — the probability assigned to outcome $x_i$ .
$x_i$ — one specific outcome, indexed by $i$ , ranging over all possible outcomes.

Play

E[X] = 2.50

Reshape the distribution and watch the expected value — the long-run average outcome — track the shift:

E[X] = \sum_i P(x_i)\, x_i

Worked example

With $P = (0.1, 0.2, 0.3, 0.4)$ over outcomes worth $(1,2,3,4)$ :

Weight each outcome by its probability
$0.1(1) = 0.1, \quad 0.2(2) = 0.4, \quad 0.3(3) = 0.9, \quad 0.4(4) = 1.6$
Add them up for the expected value
$E[X] = 0.1+0.4+0.9+1.6 = 3.0$

Checkpoint

Reshape the distribution until E[X] reads (approximately) 2.0.

E[X] = 2.50

Drag a bar to try it

Summary

\sum_i P(x_i) = 1, \qquad E[X] = \sum_i P(x_i)\, x_i

A distribution is just a set of non-negative shares that add up to one. The expected value is what you'd average over many draws from it.