Probability: Formula, Sample Space and Examples

Probability measures how likely an event is, expressed as a number between 0 and 1. A probability of 0 means impossible and 1 means certain. It is introduced in Year 8 (SMP) and developed further in senior secondary. Probability is also the foundation of the statistical thinking used in data science and artificial intelligence.

The Probability Formula

Where P(A) is the probability of event A, n(A) is the number of favourable outcomes, and n(S) is the number of all possible outcomes (the sample space).

Sample Space and Sample Points

The sample space (S) is the set of every possible outcome, and each member is a sample point. The idea comes straight from sets.

Look at the two coins: many children answer n(S) = 3 because they treat HT and TH as the same. The order differs, so the answer is 4. This classic error then contaminates every question that follows.

Worked Examples

Example 1 — one die. What is the probability of rolling an even number?

Even outcomes = {2, 4, 6}, so n(A) = 3 and n(S) = 6.

P(even) = 3/6 = 1/2

Example 2 — marbles. A bag holds 4 red and 6 blue marbles. What is the probability of drawing a red one?

n(A) = 4, n(S) = 4 + 6 = 10 → P(red) = 4/10 = 2/5

Example 3 — two dice. What is the probability that the two numbers add to 7?

Pairs totalling 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 pairs.

P = 6/36 = 1/6

Complementary Probability

The probability that an event does not happen:

If the chance of rain tomorrow is 0.3, the chance of no rain is 0.7. This often saves a great deal of work: "at least one" questions are almost always faster through the complement.

Expected Frequency

Expected frequency estimates how often an event occurs when an experiment is repeated:

Roll one die 60 times and the expected frequency of a 3 is 1/6 × 60 = 10. It is worth stressing that this is an estimate, not a guarantee. Getting 8 or 13 in practice is perfectly normal.

Experimental vs Theoretical Probability

Theoretical probability comes from the formula. Experimental probability comes from actually running the trial: occurrences divided by attempts. Toss a coin ten times and you might get seven heads — experimentally 0.7, theoretically 0.5. The more trials, the closer they converge. Explaining this to a child also plants the intuition that small samples cannot be trusted, which is useful far beyond maths lessons.

The Mistakes That Actually Cost Marks

1. Miscounting the sample space. This is the number one cause. If n(S) is wrong, every answer is wrong even with the right formula. For two dice, n(S) = 36, not 12.

2. Giving a probability greater than 1. Probability always lies between 0 and 1. A result like 7/6 is a certain sign of a slip, and a child can spot it without help.

3. Swapping n(A) and n(S) in the fraction, which inverts the answer.

4. Believing past outcomes affect the next one. A coin that has landed heads five times running still has a 1/2 chance on the sixth toss. Coins have no memory.

5. Forgetting to simplify the fraction — which points back to fractions, not to probability.