Probability Basics

The Probability Scale

Probability is always a number between 0 and 1 (inclusive).

• $P = 0$: the event is impossible
• $P = 1$: the event is certain
• $P = 0.5$: the event is equally likely to happen or not

Probabilities can be written as fractions, decimals or percentages.

Theoretical Probability

When all outcomes are equally likely:

$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of possible outcomes}}$

Example: A fair six-sided die is rolled. The probability of getting a 4:

$P(4) = \frac{1}{6}$

Example: A bag contains 3 red, 5 blue and 2 green marbles. The probability of picking a blue marble:

$P(\text{blue}) = \frac{5}{10} = \frac{1}{2}$

Experimental Probability (Relative Frequency)

When outcomes are not equally likely, or when we use data from experiments:

$P(\text{event}) = \frac{\text{number of times the event occurred}}{\text{total number of trials}}$

Example: A biased coin is flipped 200 times and lands on heads 130 times.

$P(\text{heads}) = \frac{130}{200} = 0.65$

The more trials you carry out, the more reliable the experimental probability becomes.

The Complement

The probability that an event does not happen is called the complement. If the probability of event $A$ is $P(A)$, then:

$P(\text{not } A) = 1 - P(A)$

Example: The probability of rain tomorrow is 0.3. The probability of no rain is:

$P(\text{no rain}) = 1 - 0.3 = 0.7$

Sample Space Diagrams

A sample space diagram lists all possible outcomes of a combined event. It is particularly useful when two events are combined (e.g., rolling two dice, spinning two spinners).

Example: Two fair coins are tossed. The sample space is:

	Heads	Tails
Heads	HH	HT
Tails	TH	TT

There are 4 equally likely outcomes.

$P(\text{two heads}) = \frac{1}{4}$

$P(\text{exactly one head}) = \frac{2}{4} = \frac{1}{2}$

Sample Space for Two Dice

When two dice are rolled and scores are added, there are $6 \times 6 = 36$ possible outcomes. A two-way table helps organise these.

For example, $P(\text{total} = 7)$: the combinations that give 7 are $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ — that is 6 outcomes.

$P(\text{total} = 7) = \frac{6}{36} = \frac{1}{6}$

Mutually Exclusive Events

Two events are mutually exclusive if they cannot both happen at the same time.

Example: When rolling a die, getting a 3 and getting a 5 are mutually exclusive — you cannot roll both at once.

For mutually exclusive events:

$P(A \text{ or } B) = P(A) + P(B)$

Example: $P(3 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}$

Exhaustive Events

A set of events is exhaustive if they cover all possible outcomes. The probabilities of all exhaustive, mutually exclusive events must sum to 1.

$P(A) + P(B) + P(C) + \ldots = 1$

This is useful for finding a missing probability. If $P(\text{red}) = 0.3$ and $P(\text{blue}) = 0.5$ and the only other option is green:

$P(\text{green}) = 1 - 0.3 - 0.5 = 0.2$

Expected Outcomes

The expected number of times an event occurs in $n$ trials:

$\text{Expected frequency} = P(\text{event}) \times n$

Example: A fair die is rolled 300 times. Expected number of sixes:

$\frac{1}{6} \times 300 = 50$

Note: this is a theoretical expectation. In practice, you might not get exactly 50 sixes.

Tip 1: Write Probabilities as Simplified Fractions

Unless the question asks for a decimal or percentage, give probabilities as fractions in their simplest form. This is the standard approach in GCSE mark schemes.

Tip 2: Use $1 - P$ to Find the Complement

If a question asks for the probability of something not happening, it is usually easier to find the probability of it happening and subtract from 1.

Tip 3: Draw Sample Space Diagrams

For combined events (two dice, two spinners, etc.), always draw the full sample space table. It prevents counting errors and makes it easy to spot the outcomes you need.

Tip 4: Check That Probabilities Sum to 1

If you are given probabilities for all outcomes, add them up. They should equal 1. If they don't, either the question provides incomplete information or there is an error.

Tip 5: Distinguish Theoretical from Experimental

Theoretical probability uses equally likely outcomes and logic. Experimental probability uses observed data. Questions may ask you to compare the two — the experimental probability approaches the theoretical probability as the number of trials increases.