Probability Basics: KS3 Maths Guide with Worked Examples

Probability is the measure of how likely an event is to happen. It is always a number between 0 (impossible) and 1 (certain), and it can be written as a fraction, decimal, or percentage. KS3 probability — taught from Year 7 — covers the probability scale, calculating simple theoretical probabilities, listing all possible outcomes, and working with complementary events.

The probability scale

All probabilities sit on a number line from 0 to 1:

Value | Meaning | Example event
0 | Impossible — cannot happen | Rolling a 7 on a standard six-sided die
0 to 0.5 | Unlikely — less likely than not | Rolling a 6 on a fair die (probability 1/6)
0.5 | Even chance — equally likely or unlikely | Flipping heads on a fair coin
0.5 to 1 | Likely — more likely than not | Rolling a number less than 5 on a fair die (probability 4/6)
1 | Certain — must happen | Rolling a number less than 7 on a standard die

The basic probability formula

For equally likely outcomes:

P(event) = number of favourable outcomes ÷ total number of possible outcomes

This only applies when every possible outcome is equally likely — a fair die, a fair coin, a well-shuffled deck.

Worked example 1 — rolling a die

A fair six-sided die is rolled once. The faces are numbered 1 to 6.

What is the probability of rolling a 4?

P(4) = 1 ÷ 6 = 1/6

What is the probability of rolling an even number?

Even numbers on a standard die: 2, 4, 6 — that is 3 favourable outcomes.

P(even) = 3 ÷ 6 = 1/2

Worked example 2 — choosing from a bag of counters

A bag contains 5 red counters, 3 blue counters, and 2 green counters.

Total counters = 5 + 3 + 2 = 10.

P(red) = 5/10 = 1/2

P(blue) = 3/10

P(green) = 2/10 = 1/5

P(not red) = (3 + 2)/10 = 5/10 = 1/2

Listing all possible outcomes

When a probability problem involves two or more events (such as flipping two coins, or rolling two dice), listing every possible combination prevents you from missing outcomes. This complete list is called the sample space.

Sample space table — two coins

Coin 1 | Coin 2 | Outcome
Heads | Heads | HH
Heads | Tails | HT
Tails | Heads | TH
Tails | Tails | TT

Total outcomes = 4.

P(two heads) = 1/4

P(exactly one head) = 2/4 = 1/2 (HT and TH)

P(at least one tail) = 3/4 (HT, TH, TT)

Sample space table — two dice

When two fair dice are rolled, there are 6 × 6 = 36 equally likely outcomes. A sample space table lists all 36 pairs.

Worked example 3: What is the probability that two fair dice give a total of 7?

Pairs summing to 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — that is 6 pairs.

P(sum = 7) = 6/36 = 1/6

Worked example 4: What is the probability that the total is greater than 9?

Pairs: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6) — that is 6 pairs.

P(sum > 9) = 6/36 = 1/6

Complementary events

The complement of an event A (written A' or "not A") is the event that A does not happen. Since the event either happens or it does not:

P(A) + P(not A) = 1

Rearranging: P(not A) = 1 − P(A)

Worked example 5 — complementary events

The probability that it rains on a given day in November is 0.65. What is the probability that it does not rain?

P(not rain) = 1 − 0.65 = 0.35

Worked example 6 — counters (revisited)

In the bag from Example 2 (5 red, 3 blue, 2 green), what is the probability of not picking a green counter?

Method 1 (direct): P(not green) = 8/10 = 4/5

Method 2 (complement): P(not green) = 1 − P(green) = 1 − 2/10 = 8/10 = 4/5 ✓

Both methods agree — always a useful check.

Experimental probability

Theoretical probability uses equally likely outcomes. Experimental probability (or relative frequency) is calculated from the results of an actual experiment:

Experimental probability = number of times event occurred ÷ total number of trials

The more trials you carry out, the closer the experimental probability gets to the theoretical probability. This is the law of large numbers.

Example: A drawing pin is dropped 200 times and lands point-up 84 times.

P(point up) = 84/200 = 0.42

We cannot calculate the theoretical probability of a drawing pin landing point-up from equally likely outcomes, so the experimental value is our best estimate.

How probability fits the KS3 national curriculum

The Department for Education's KS3 mathematics programme of study requires pupils to "record, describe and analyse the frequency of outcomes of simple probability experiments using tables and frequency trees," and to "apply ideas of randomness, fairness and equally likely events to calculate expected outcomes." The AQA GCSE Mathematics specification covers the probability scale, calculating basic probabilities, and complementary events as core assessment topics for Year 7 to Year 9 pupils.

Common mistakes

Mistake 1 — Writing a probability greater than 1. Probability is always between 0 and 1 inclusive. P = 7/5 is impossible. If you get a fraction greater than 1 (or a percentage greater than 100 %), recount your outcomes.

Mistake 2 — Confusing relative frequency with expected frequency. Relative frequency (probability) tells you how often an outcome occurs per trial. Expected frequency = probability × number of trials. They are different things with different units.

Mistake 3 — Forgetting all outcomes when listing a sample space. With two dice, it is easy to miss combinations or double-count. Always draw the full table to be sure.

Mistake 4 — Not simplifying fractions. 4/12 must be simplified to 1/3. Leaving it unsimplified costs a mark in GCSE exams and may cause confusion.

Frequently asked questions

What is the difference between theoretical and experimental probability?

Theoretical probability is calculated by reasoning about equally likely outcomes before carrying out any experiment. Experimental probability (relative frequency) is measured by actually performing the experiment and recording how often each outcome occurs. For a fair coin, the theoretical probability of heads is 1/2. If you flip the coin 100 times and get 53 heads, the experimental probability is 53/100 = 0.53. With more trials, experimental probability converges towards the theoretical value.

Can two events have the same probability?

Yes. When several outcomes are equally likely, each has the same probability. For example, on a fair six-sided die, each face has probability 1/6. In a bag with 4 identical red and 4 identical blue counters, P(red) = P(blue) = 1/2. Probabilities for different events can also happen to be equal by coincidence — for instance, on a fair die, P(rolling a 3) = P(rolling a number > 5), both equal 1/6.

Why do all the probabilities of all possible outcomes add up to 1?

Because one of the possible outcomes must happen — it is certain that some outcome occurs. The sum of probabilities of an exhaustive set of mutually exclusive outcomes is always 1. For a fair die: P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 6 × 1/6 = 1. This is a useful check: if your probabilities for all outcomes do not add to 1, you have made an error.

How do I know when to list outcomes versus using a formula?

Use the formula P = favourable outcomes ÷ total outcomes when you can easily count both quantities. List all outcomes in a sample space table when the problem involves two or more combined events (two dice, two spinners, two draws) and the total number of outcomes is not immediately obvious. The table makes it easy to count correctly without missing any combination.

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