Probability is not a machine for predicting one future event with certainty. It is a mathematical model for describing uncertainty across possible outcomes.
A single coin toss can still surprise us. A probability model does not promise that heads must appear after tails, or that five losses make a win “due”. Instead, it gives a disciplined way to describe possibilities, compare risks, and reason about repeated trials without pretending uncertainty has disappeared.
One-sentence answer
Probability measures how strongly a model supports an event, from impossible to certain, while keeping the uncertainty visible.
The scale from 0 to 1
Probabilities are often written as fractions, decimals or percentages.
0 = impossible
1 = certain
0.5 = equally balanced in a simple two-outcome modelA probability of 0.2 is the same as 20% or 1/5. These are three representations of the same quantity.
Model first, calculation second
Before calculating probability, identify the model. What outcomes are possible? Are they equally likely? Is the process with replacement or without replacement? Does one event change the next event?
This is why the familiar formula
P(event) = favourable outcomes / total outcomesis safe only when the listed outcomes are equally likely. If a spinner has unequal sectors, simply counting colours can be wrong because the sectors do not carry equal probability.
Worked example 1: a fair die
A fair six-sided die is rolled. Find the probability of obtaining an even number.
The equally likely outcomes are 1, 2, 3, 4, 5 and 6. The favourable outcomes are 2, 4 and 6.
P(even) = 3/6 = 1/2The probability is 1/2.
Worked example 2: complement
If the probability that a bus arrives within five minutes is 0.72, the probability that it does not arrive within five minutes is:
P(not within five minutes) = 1 − 0.72 = 0.28An event and its complement fill the entire model, so their probabilities add to 1.
Worked example 3: mutually exclusive events
A fair die is rolled. What is the probability of obtaining a 1 or a 6?
The events cannot happen together on one roll, so they are mutually exclusive.
P(1 or 6) = 1/6 + 1/6 = 1/3The addition works because there is no overlap to double-count.
When events overlap
If events can happen together, adding their probabilities directly may double-count the overlap.
For a card numbered 1 to 20, let A be “multiple of 2” and B be “multiple of 5”. The number 10 and 20 belong to both sets. A correct count must include them once, not twice.
This is why probability connects naturally to set language and Venn diagrams.
Independent events
Two events are independent when the occurrence of one does not change the probability of the other. Repeated tosses of a fair coin are modelled as independent.
For two fair coin tosses:
P(head then head) = 1/2 × 1/2 = 1/4Multiplication follows the path through both events.
Dependent events
If a bag contains 3 red and 2 blue counters and one counter is removed without replacement, the second draw depends on the first because the contents have changed.
If the first counter is red, only 2 red and 2 blue counters remain. The probability structure must therefore update.
The gambler’s fallacy
If a fair coin lands heads five times in a row, tails is not mathematically “due” on the sixth toss. Under the independent fair-coin model, the sixth toss is still 1/2 heads and 1/2 tails.
The previous sequence is unusual, but it does not change the probability of the next independent toss.
Expected frequency
If an event has probability 0.3 and an experiment is repeated 200 times, an expected frequency is:
200 × 0.3 = 60This does not guarantee exactly 60 occurrences. It gives the model-based long-run expectation.
Probability does not erase uncertainty
A weather forecast of 70% rain does not mean it will rain for 70% of the day, nor that 70% of the local area must receive rain. The interpretation depends on the forecasting model and definition used.
Good probability reasoning therefore separates a numerical probability from the assumptions that produced it.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Counts outcomes that are not equally likely | Model selection | Represent actual probabilities or areas |
| Adds overlapping events directly | Double-counting | Identify intersection first |
| Keeps probability unchanged after no-replacement draw | Dependence | Update the sample space |
| Says a result is “due” | Independence misconception | Separate sequence surprise from next-trial probability |
| Expects observed frequency to equal theoretical probability exactly | Random variation | Compare long-run tendency, not exact short-run matching |
Practice
- A fair die is rolled. Find P(number greater than 4).
- If P(A) = 0.63, find P(not A).
- A fair coin is tossed twice. Find P(two tails).
- An event has probability 0.18. Find its expected frequency in 500 trials.
- Explain why five consecutive heads do not make tails more likely on the next fair independent toss.
Answers
1. 2/6 = 1/3. 2. 0.37. 3. 1/4. 4. 90. 5. Each toss is independent; the next toss remains 1/2 for tails under the fair-coin model.
Connected routes
Continue to Sample Space Diagrams: Building Outcomes Systematically, Tree Diagrams Without Double-Counting, and Experimental and Theoretical Probability. Return to the Mathematics Learning Hub for the wider Secondary Mathematics route.