Before the method: the idea behind it
Putting numbers on uncertainty
Calculus is done. This topic starts something different: describing situations whose outcome you cannot predict, but whose behaviour over many repetitions is remarkably predictable. A single die roll is a mystery. Ten thousand die rolls average almost exactly 3.5, every time. Random variables are the tool that makes that second fact precise, and they lead directly to the statistical inference you will meet in Term 3.
Part one
A variable whose value is decided by chance
A random variable is a rule that attaches a number to each outcome of a random process.
Roll a die and let X be the number showing. Toss three coins and let X be how many land heads. Count the cars passing in a minute. In every case the value of X is not fixed: it is settled by chance, and different repetitions give different values. What is fixed is the list of possible values and how likely each one is. That list is called the probability distribution, and it is the complete description of the variable.
Because we can list the possible values one at a time, X is called discrete. Later this year you will meet continuous random variables, where the possible values fill an entire interval and listing them is impossible.
Two rules govern every distribution: no probability can be negative, and the probabilities must total exactly 1, because something must happen.
Part two
Why long-run behaviour is predictable
Roll a die a few times and the results look chaotic. Roll it thousands of times and a pattern appears with almost mechanical reliability.
Press the button below to simulate rolls. Watch how the bars, which start jagged and uneven, settle towards the flat dashed line of the true distribution. This is why probability is useful: it does not predict the next roll, but it predicts the shape of many rolls with great accuracy.
With few rolls the bars are uneven and the average jumps around. Keep rolling and both settle down.
That settling value of the average has a name: the expected value, written E(X) or μ. It is not the value you expect to see on any one roll (3.5 is not even possible), it is the value the average of many trials closes in on. You compute it by weighting each outcome by how likely it is.
Part three
The balance point, and how far things spread
There is a physical picture that makes the mean concrete: put a weight at each possible value, with the weight equal to its probability. The mean is where the plank balances.
Drag the probabilities below and watch the pivot slide. Load more probability onto the high values and the balance point moves right. It is exactly the centre of mass, which is why the mean is called a measure of location.
Probabilities are rescaled to total 1 automatically. The teal bar shows one standard deviation either side of the mean.
The mean tells you where the distribution sits, but not how spread out it is. Two variables can share a mean and behave completely differently: a payout that is always $10 and a coin flip paying $0 or $20 both have mean $10, but only one carries risk.
Spread is measured by the variance: the average squared distance from the mean. Squaring stops positive and negative deviations from cancelling. Because squaring also changes the units, we usually take the square root and quote the standard deviation instead, which is back in the original units.
That formula is the practical one: work out the average of the squares, then subtract the square of the average. It is much quicker than computing every deviation individually, and it is the version you should use in exams.
Where this shows up in real life
Insurance and expected value
An insurer cannot know whether your house will burn down, but across a million policies the expected payout is highly predictable. Premiums are set as the expected payout plus a margin. The entire industry is an application of E(X).
Why casinos always win
Every casino game has a negative expected value for the player, usually just a few cents per dollar. A single player might win big, but the house plays millions of rounds, and over that many trials the average is essentially guaranteed. Same mathematics as the die simulation above.
Before you start
What to reach for
| The question asks for… | Use |
|---|---|
| an unknown probability in a table | All probabilities sum to 1 |
| the mean, expected value, or long-run average | E(X) = Σ x P(X = x) |
| variance or standard deviation | Var(X) = E(X²) − [E(X)]², then square root |
| the mean or variance of aX + b | E(aX+b) = aE(X)+b, Var(aX+b) = a²Var(X) |
| whether a game is fair | Compute E(profit); fair means zero |
Next: the working method, twenty practice questions, ten harder ones, and the ClassPad guide for computing all of this from a table in seconds.