Year 12: Probability

Probability at this level moves beyond counting events into modelling with distributions and density functions. The main challenge is deciding what the random variable means before you start calculating.

Define the random variable clearly, then write down the relevant support, conditions and total probability requirement before computing anything else.

Probability Tutorials

What to master

Using conditional probability carefully and recognising when dependence matters.
Reading discrete probability distributions and checking they are valid.
Calculating probabilities, expected values and related statistics from a distribution.
Working with simple probability density functions over continuous intervals.

Exam habit: when a density function is involved, check the interval first. Integrating outside the supported range is a common source of lost marks.

How the topic changes in Year 12

Earlier probability work often focuses on listing outcomes and counting favourable cases. In Year 12, that still matters, but now you also model uncertainty with random variables and distributions. That means every calculation should begin with interpretation. What does the variable represent? What values can it take? Are the outcomes discrete or continuous? Once those questions are answered, the formulas become much easier to choose correctly.

Worked example

Problem: A random variable

X

has

P (X = 0) = 0.2

P (X = 1) = 0.5

and

P (X = 2) = 0.3

. Find

P (X \geq 1)

and the expected value

E (X)

For $P (X \geq 1)$ , add the probabilities for 1 and 2: $0.5 + 0.3 = 0.8$ .
Now calculate the expected value: $E (X) = 0 (0.2) + 1 (0.5) + 2 (0.3)$ .
This gives $E (X) = 0 + 0.5 + 0.6 = 1.1$ .

Answer:

P (X \geq 1) = 0.8

and

E (X) = 1.1

Common mistakes

Using counting formulas when the question is really about distributions.
Failing to check whether the probabilities in a table actually add to 1.
For density functions, forgetting that probability comes from area, not point values.
Interpreting expected value as the most likely single outcome.

In revision, try to describe every probability object in words before calculating. If you can say what the random variable measures, what outcomes are possible, and whether the setting is discrete or continuous, the correct method usually becomes obvious. That description step prevents many formula mistakes before they happen.

Revision focus

Probability work becomes far more reliable when you define the object of the question before calculating. Is the problem about ordered outcomes, unordered selections, a conditional event, a discrete random variable, or a continuous density? That decision controls the method. In many exam questions, the major error is not arithmetic. It is using a correct formula for the wrong kind of probability situation.

A second useful habit is to interpret the result immediately after it is found. Decide whether the answer is a count, a probability, an average outcome, or a spread measure. In continuous settings, remember that probabilities come from areas over intervals, not from single points. In discrete settings, remember that the total probability must still add to one. Those interpretation checks are often the fastest way to catch an error before it reaches the final line.