Year 12 Probability: Discrete probability distributions

A discrete probability distribution lists the possible values of a random variable and the probability attached to each value.

Before calculating anything, check two things: every probability is between 0 and 1, and the total of all probabilities is 1.

How to read one

Identify the random variable and what it measures.
List the possible values of the variable.
Attach a probability to each value.
Check that the full distribution is valid.

Worked example

Problem: A random variable

X

takes the values 0, 1 and 2 with probabilities

k

2 k

and

3 k

respectively. Find

k

and then find

P (X \geq 1)

The probabilities in a discrete distribution must add to 1: $k + 2 k + 3 k = 1$ .
So $6 k = 1$ , which gives $k = 1 / 6$ .
Now calculate $P (X \geq 1)$ : $P (X = 1) + P (X = 2) = 2 k + 3 k = 5 k$ .
Substitute $k = 1 / 6$ : $P (X \geq 1) = 5 / 6$ .

Answer:

k = 1 / 6

and

P (X \geq 1) = 5 / 6

Why validity checks matter

A table of numbers is not automatically a probability distribution. Every value must be between 0 and 1, and the total must be exactly 1. Those checks are not decoration. They are often the first part of a question, especially when one of the probabilities contains an unknown constant.

Worked example 2

Problem: Let

P (X = 1) = 0.2

P (X = 2) = 0.3

and

P (X = 3) = 0.5

. Find

P (X < 3)

The event $X < 3$ means $X = 1$ or $X = 2$ .
Add those probabilities: $0.2 + 0.3 = 0.5$ .

Answer:

P (X < 3) = 0.5

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.

In probability, the final line should always answer the right event, not just display a calculation. After computing, read the question again and confirm that your sample space, condition, or random-variable interpretation still matches what was asked. That last comparison catches a surprising number of near-miss errors.