A theoretical probability is a probability based on counting the number of desired outcomes from a sample space where all individual outcomes are equally likely.

Checking and securing displaying outcomes

Maths

In this unit, pupils use their knowledge of calculating theoretical probabilities and extend this understanding to calculating conditional probabilities using all of the representations encountered in the prior unit.

In this unit, pupils use their knowledge of calculating theoretical probabilities and extend it to calculate conditional probabilities.

Conditional probability

Two or more events are mutually exclusive if they share no common outcome.

A set of events are exhaustive if at least one of them has to occur whenever the experiment is carried out.

Checking and securing exhaustive events

The probability that an event will occur is the proportion of times the event is expected to happen in a suitably large experiment.

Probabilities involving algebra

The frequency is the number of times an event occurs; or the number of individuals (people, animals etc.) with some specific property.

Tree diagrams are a representation used to model statistical/probability questions. Branches represent different possible events or outcomes.

Frequency trees

Checking and securing calculating probabilities from tables

Two or more events are are mutually exclusive if they share no common outcome.

Checking and securing calculating probabilities from diagrams

The frequency of an outcome from an experiment can be used to produce an experimental probability. This is sometimes referred to as relative frequency.

A trial or experiment is fair if each outcome has an equal chance of happening. Each outcome is said to be equally likely.

If there is bias, in a trial or experiment, the outcomes are not equally likely.

Experimental probability

An experiment is a repetition of a trial multiple times in order to observe how often each outcome occurs.

Experimental vs theoretical probability

Venn diagrams are a representation used to model statistical/probability questions. Commonly circles are used to represent events.

Set notation

Conditional probability is the probability of an outcome occurring given that another event has occurred.

Mutually exclusive events have no outcomes in common.

Event A is independent of event B if the probability of event A occurring is not affected by whether or not event B occurs.

Conditional probability in a two-way table

Conditional probability in a Venn diagram

Each branch of a probability tree shows a possible outcome from an event or from a stage of a trial, along with the probability of that outcome happening.

Conditional probability in a tree diagram

Algebra in tree and Venn diagrams

An outcome is a result of a trial (e.g. getting heads when flipping a coin once or getting two heads when flipping a coin twice).

A sample space is all the possible outcomes of a trial. A sample space diagram is a systematic way of producing a sample space.

An event is a subset of a sample space. i.e. An outcome or set of outcomes that may occur from a trial (e.g. flipping a coin twice and getting the same result each time).

Constructing three event Venn diagrams

Probabilities in three event Venn diagrams

Comparing multiple representations to calculate conditional probabilities

When listing outcomes systematically, they are listed in such a way as to ensure all outcomes are recorded.

Conditional probability

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These lessons will be removed by end of Summer Term 2025.