Introduction to probability.
Describe chance, define events and calculate probabilities from equally likely outcomes.
Use the 0–1 scale and P(A) notation · count outcomes · calculate complements · interpret tables and graphs
Place the chance between 0 and 1
Chance describes uncertainty before an outcome is known. Some events cannot happen, some must happen, and many lie between those extremes. Probability assigns a number to that degree of likelihood.
Zero means impossible and 1 means certain. Halfway between them is 12, an even chance. Fractions, decimals and percentages can name the same probability, so connect the number to its meaning on the scale rather than treating it as an isolated calculation.
Probability measures how likely an event is. Every probability lies between 0 and 1, inclusive. Larger values mean more likely events.
P(A′) = 1 − 0.60 = 0.40
Pause and explainDoes a probability of 0.8 mean an event is certain?
No. It is likely, corresponding to 80%, but there is still a 20% chance that it does not occur.
Name the event before counting
Before calculating a chance, describe the random action and the event of interest. Rolling a die is an experiment; landing on 4 is an outcome; landing on an even number is an event containing several outcomes.
List the full sample space and identify the favourable outcomes inside it. Counting favourable outcomes over all outcomes works when those elementary outcomes are equally likely. A biased die or an unequal-sector spinner requires probabilities that reflect that unequal chance.
An outcome is one possible result, such as rolling a 4. An event is a collection of outcomes, such as rolling an even number. The sample space contains all possible outcomes.
A fair six-sided die is rolled. Find P(prime number).
The sample space is {1, 2, 3, 4, 5, 6}. The prime outcomes are 2, 3 and 5.
One is not prime. The event has three favourable outcomes out of six equally likely possibilities.
Pause and explainWhy can you not always divide by the number of colours on a spinner?
The colours may occupy different-sized sectors, so the outcomes may not be equally likely.
Use the complement for ‘not’
For any event, either it happens or it does not. Those two possibilities cover everything and cannot occur together. This is the simple reason their probabilities sum to 1.
The complement notation A′ means “not A”. It may contain several different outcomes, not just one alternative. Identify what is outside the event, then subtract its probability from 1 when that is easier than counting the complement directly.
A′ is the complement of A: every outcome where A does not happen. Exactly one of A and A′ happens, so their probabilities sum to 1.
P(B) = 0.8. Find P(B′).
A bag has 7 red counters and 5 blue counters. One is chosen at random. Find P(not red).
Here, ‘not red’ means blue because the bag contains only those two colours.
Pause and explainIf A means rolling a 6, does A′ mean rolling a 5 only?
No. It includes every result other than 6: 1, 2, 3, 4 and 5.
Read frequencies from a table or graph
A table or frequency graph can describe a population from which one person or object is chosen at random. Each frequency counts how many possible selections belong to a category.
The denominator must be the number of selectable individuals in the stated group, not the number of categories or bars. Add relevant frequencies for the numerator, taking care that overlapping Venn regions are not counted twice.
| Activity | Sport | Music | Art |
|---|---|---|---|
| Frequency | 12 | 10 | 8 |
Choose one of these students at random. Find P(music) and P(not art).
P(not art) = 2230 = 1115
The denominator is the total number of students, not the number of categories. For a frequency graph, add its frequencies first. In a Venn diagram, count the relevant regions, including any overlap only once.
Pause and explainIf three categories contain 4, 6 and 10 people, what is the denominator when selecting one person from all of them?
20, the total number of people, not 3, the number of categories.
Your turn. State the event and its complement.
Name the event in words and identify the full collection being chosen from. Check whether its outcomes are equally likely before using a simple counting fraction.
Use complements to simplify “not” questions, and verify that your final probability lies between 0 and 1. A correct fraction should be simplified without changing its meaning.
A probability is written as 1.2. Explain the error.
Reveal worked answer
A fair die is rolled. Find P(number greater than 4).
Reveal worked answer
P(A) = 0.37. Find P(A′).
Reveal worked answer
A bag has 6 green, 3 yellow and 1 white counter. One is selected at random. Find P(not yellow).
Reveal worked answer
The activity table above describes 30 students. Find the probability that a randomly chosen student prefers sport or art.