Probability Basics and Rules
Probability measures how likely an event is. Start by identifying the possible outcomes, then use a suitable model to calculate and check the chance of the event you want.
Foundation & Higher. You will use fractions, decimals and percentages. Review Fractions, Decimals & Percentages if you need to move between these forms.
Outcomes, events and the probability scale
A trial is one performance of an experiment, such as rolling a die. An outcome is one possible result. An event is a collection of outcomes: “roll an even number” means 2, 4 or 6.
Write P(A) for the probability of event A. A probability is between 0 and 1 inclusive, and has no units. You can also write it as a fraction or a percentage.
| Likelihood | Probability |
|---|---|
| Impossible | 0 |
| Unlikely | Between 0 and 0.5 |
| Even chance | 0.5 = 1/2 = 50% |
| Likely | Between 0.5 and 1 |
| Certain | 1 = 100% |
“Unlikely” does not mean impossible. A likely event can still fail to happen on the next trial.
Theoretical probability and equally likely outcomes
For a model with equally likely outcomes, probability = number of favourable outcomes ÷ total number of possible outcomes. Count individual outcomes, not just the names of categories.
A fair six-sided die has six equally likely faces. A random selection is made by chance; that alone does not make every category equally likely. A fair spinner divided into unequal-sized sectors lands in larger sectors more often.
Worked example — count the objects
A bag contains 5 red, 3 blue and 2 green counters. It is mixed, and each counter is equally likely to be selected. There are 10 counters, so P(blue) = 3/10 = 0.3 = 30%.
There are three colours, but their probabilities are not all 1/3: the colours contain different numbers of counters.
Worked example — count several favourable outcomes
On a fair six-sided die, the prime numbers are 2, 3 and 5. P(prime) = 3/6 = 1/2. The number 1 is not prime.
Always state or check the model: a biased die cannot be treated as six equally likely faces without further information.
Complements: an event and its opposite
The complement of A means “not A”. Exactly one of A or not A happens on each trial, so P(not A) = 1 − P(A).
Worked example — use the complement
The probability that a parcel arrives late is 0.18. The probability that it is not late is 1 − 0.18 = 0.82. “Not late” includes early and on-time arrivals.
Check: 0.18 + 0.82 = 1. Do not confuse “not late” with a narrower event such as “early”.
Mutually exclusive and exhaustive events
Mutually exclusive events cannot happen together in the same trial. An exhaustive collection covers every possible outcome. These are different properties.
On one die roll, “1”, “2” and “3” are mutually exclusive but not exhaustive. “Even” and “odd” are both mutually exclusive and exhaustive. “Even” and “greater than 3” overlap at 4 and 6, so they are not mutually exclusive.
For mutually exclusive events, P(A or B) = P(A) + P(B). For a collection that is also exhaustive, the probabilities add to 1. If events overlap, adding their probabilities counts the overlap twice; Venn diagrams will show how to correct this.
Worked example — add disjoint possibilities
Using the bag of 5 red, 3 blue and 2 green counters, P(red or green) = 5/10 + 2/10 = 7/10. A single counter cannot have both colours in this model.
| Outcome | Probability |
|---|---|
| Red | 0.45 |
| Blue | 0.30 |
| Yellow | ? |
If these are the only outcomes, P(yellow) = 1 − 0.45 − 0.30 = 0.25.
Find an unknown probability
Unknown probabilities still obey the total-of-1 rule. Write an equation only after checking that your listed outcomes are exhaustive and mutually exclusive.
Worked example — use a simple equation
A spinner has three possible colours with probabilities x, 2x and 0.4. Then x + 2x + 0.4 = 1, so 3x = 0.6 and x = 0.2.
The probabilities are 0.2, 0.4 and 0.4. Each is between 0 and 1, and their sum is 1.
Think Like a Mathematician
A student says “It will either rain or not rain, so the probability of rain is 1/2.” Two possible categories need not be equally likely. What information would justify a numerical probability?
Common Mistakes
Counting categories instead of equally likely objects: three colours do not automatically mean three equal chances.
Adding overlapping events: “or” includes the overlap, but it must be counted once.
Invalid values: a probability cannot be negative or greater than 1; 70% means 0.7, not 70.
Practice
1. Write an even chance as a fraction, a decimal and a percentage.
1/2, 0.5 and 50% are the same probability.
2. Which values could be probabilities: −0.1, 0, 3/5, 1 and 1.2?
0, 3/5 and 1. The valid range includes both endpoints, 0 and 1.
3. A mixed bag contains 4 white and 6 black counters, each equally likely to be selected. Find P(white).
There are 10 counters. P(white) = 4/10 = 2/5.
4. A fair six-sided die is rolled. Find P(number greater than 4).
The favourable outcomes are 5 and 6: 2/6 = 1/3.
5. P(a train is late) = 0.27. Find P(the train is not late).
1 − 0.27 = 0.73.
6. A game has only win, draw and lose outcomes. P(win) = 0.35 and P(draw) = 0.20. Find P(lose).
1 − 0.35 − 0.20 = 0.45.
7. On one fair die roll, are “odd” and “6” mutually exclusive? Are they exhaustive? Find P(odd or 6).
They are mutually exclusive, but not exhaustive: 2 and 4 are missing. P(odd or 6) = 3/6 + 1/6 = 2/3.
8. Three exhaustive, mutually exclusive outcomes have probabilities x, 3x and 0.2. Find x.
x + 3x + 0.2 = 1, so 4x = 0.8 and x = 0.2. The probabilities 0.2, 0.6 and 0.2 total 1.
9. A spinner has four colours. One colour fills half the circle. Explain why you cannot assign probability 1/4 to every colour.
The colour regions are not equally sized. In an ideal fair spin, the half-circle colour has probability 1/2, not 1/4.
Frequently Asked Questions
Must I simplify a probability fraction?
Equivalent fractions represent the same chance, but a simplified exact fraction is usually the clearest final answer unless another form is requested.
Does probability 0.8 guarantee eight successes in ten trials?
No. It describes the chance on each suitable trial. Expected counts and actual results are different; the next lesson explains this.