Revision notes

Every Probability topic condensed to one page: the facts to know, the mistakes to avoid, and the formulae. Use it as a last-minute check, or print it.

Formulae for this unit

Probability of an outcome $\dfrac{\text{favourable outcomes}}{\text{total outcomes}}$
Expected number of occurrences $\text{probability} \times \text{number of trials}$
Probability of two independent events $P(A \text{ and } B) = P(A) \times P(B)$

Basic Probability & Sample Spaces

Probability is the number of favourable outcomes divided by the total number of equally likely outcomes.
Every probability is between 0 (impossible) and 1 (certain).
The probabilities of all possible outcomes add up to 1.
P(not A) equals 1 minus P(A).

Watch out: Writing a probability as a raw count, like "5 counters", instead of as a fraction of the total number of outcomes.

The Probability Scale

0 means impossible, 1 means certain, and 0.5 means an even chance.
Write probabilities as fractions, decimals or percentages, never as ratios or as 1 in 5.
Words like unlikely and likely map to positions on the 0 to 1 scale.
An event and its opposite sit the same distance from each end of the scale.

Watch out: Describing a small but non-zero probability, such as 0.1, as "impossible" instead of "unlikely", since impossible only applies to a probability of exactly 0.

Relative Frequency

Relative frequency is the number of times something happened divided by the number of trials.
It is an estimate of probability from an experiment, also called experimental probability.
The more trials you do, the closer relative frequency gets to the true probability.
Expected frequency is probability multiplied by the number of trials.

Watch out: Dividing the total number of trials by the frequency of the event, instead of the event's frequency by the total number of trials.

Mutually Exclusive Events

Mutually exclusive events cannot both happen at the same time.
For mutually exclusive events, P(A or B) equals P(A) plus P(B).
If a set of outcomes is exhaustive and mutually exclusive, their probabilities sum to 1.
Rolling a 2 and rolling an odd number on one die are mutually exclusive.

Watch out: Assuming two events are mutually exclusive without checking whether an outcome could satisfy both, for example missing that 6 is both a multiple of 3 and an even number.

Independent Events

Independent events do not affect each other probability.
For independent events, P(A and B) equals P(A) times P(B): multiply along the branches.
Drawing with replacement keeps events independent; drawing without replacement does not.
Multiply for "and", add for "or".

Watch out: Assuming events are independent without checking whether one outcome affects the other, for example missing that picking without replacement changes the probabilities.

Tree Diagrams

Each set of branches must have probabilities that add to 1.
Multiply along the branches to get the probability of a whole path.
Add the results of the different paths that satisfy the condition.
Without replacement, the second set of probabilities changes because one item has been removed.

Watch out: Forgetting that the probabilities on branches from the same point must add up to 1 when finding a missing value.

Venn Diagrams & Set Notation

The overlap is "A and B" (the intersection); everything inside either circle is "A or B" (the union).
Anything outside both circles is in neither set.
Fill the intersection first, then the rest of each circle, then the region outside.
P(A given B) uses only the part of the diagram inside B.

Watch out: Confusing the union symbol, ∪, meaning "or" (everything in either set), with the intersection symbol, ∩, meaning "and" (only the overlap).

Conditional Probability

Conditional probability is the chance of one event given that another has already happened.
P(A given B) equals P(A and B) divided by P(B): restrict yourself to the outcomes where B is true.
"Without replacement" problems are conditional, because the second probability depends on the first.
On a tree diagram, the second set of branches shows conditional probabilities.

Watch out: Dividing by the overall total instead of the total for the given condition when finding a conditional probability from a Venn diagram or table.

Two-Way Tables

A two-way table sorts data by two categories at once, and the totals must agree both ways.
Fill in the totals first, then find the missing cells by subtraction.
Read a probability straight from the table: the matching cell over the relevant total.
Check every row and column adds to its total before you answer.

Watch out: Subtracting from the wrong row or column total when finding a missing value, instead of using the total that contains the missing cell.

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