Mutually exclusive and exhaustive events Revision Notes for AQA GCSE Statistics

Mutually exclusive and exhaustive events

What are mutually exclusive events?

Mutually exclusive events are events that cannot happen at the same time. Think of it this way - if one event occurs, the other absolutely cannot occur in the same trial or situation.

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The key to understanding mutually exclusive events is remembering that they are completely separate outcomes - there is no overlap between them whatsoever.

For example, when you roll a standard dice, you cannot get both a 3 and a 5 on the same roll. These outcomes are mutually exclusive because only one number can appear on the top face of the dice.

Similarly, when flipping a coin, you cannot get both heads and tails on a single flip - these are mutually exclusive outcomes.

Writing probabilities

When working with probability, we use specific notation to express the likelihood of events occurring.

For a standard dice with six equally likely outcomes, the probability of rolling a 6 can be written as:

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

This means there is 1 favourable outcome (rolling a 6) out of 6 possible outcomes (1, 2, 3, 4, 5, 6).

For a fair coin with two equally likely outcomes, the probability of getting heads can be written as:

$P(\text{heads}) = \frac{1}{2}$

This represents 1 favourable outcome (heads) out of 2 possible outcomes (heads or tails).

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Probability notation uses $P(\text{event})$ to represent "the probability that the event occurs". The result is always a number between 0 and 1, where 0 means impossible and 1 means certain.

Sum of probabilities (exhaustive events)

A crucial principle in probability is that all possible outcomes of an event must add up to 1. This concept relates to exhaustive events - events that cover all possible outcomes with no gaps.

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If you know the probability that something will happen, you can calculate the probability that it won't happen using the complement rule - this is one of the most useful tools in probability!

The complement rule states: $P(\text{event A doesn't happen}) = 1 - P(\text{event A happens})$

This can be written more concisely as: $P(\text{not A}) = 1 - P(A)$

This formula is extremely useful when it's easier to calculate the probability of something not happening rather than it happening.

Formula for mutually exclusive events

When two events A and B are mutually exclusive, we can use the addition law:

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

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This is sometimes called the "or" rule for mutually exclusive events. The key word here is "or" - we want to find the probability that either event A happens OR event B happens (but not both, since they're mutually exclusive).

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Worked Example: Spinner Probability

Consider a spinner with three sections: red, blue, and yellow, where:

$P(\text{red}) = 0.4$
$P(\text{blue}) = 0.2$
$P(\text{yellow}) = 0.4$

Since these are the only three possible outcomes, they are exhaustive (they add up to 1: $0.4 + 0.2 + 0.4 = 1.0$ ).

We can calculate various combinations:

$P(\text{red or yellow}) = P(\text{red}) + P(\text{yellow}) = 0.4 + 0.4 = 0.8$
$P(\text{red or blue}) = P(\text{red}) + P(\text{blue}) = 0.4 + 0.2 = 0.6$
$P(\text{blue or yellow}) = P(\text{blue}) + P(\text{yellow}) = 0.2 + 0.4 = 0.6$

Worked examples

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Worked Example 1: Finding Missing Probabilities

Consider this probability table for selecting coloured beads:

Table

Part (a): Work out the value of x

Since all probabilities must add up to 1: $P(\text{Red}) + P(\text{Green}) + P(\text{Blue}) + P(\text{Yellow}) = 1$ $0.1 + 0.25 + x + 0.3 = 1$ $0.65 + x = 1$ $x = 1 - 0.65 = 0.35$

Part (b): Work out the probability that a green bead or a yellow bead is taken

Since green and yellow are mutually exclusive events (you can't select both simultaneously): $P(\text{green or yellow}) = P(\text{green}) + P(\text{yellow})$ $P(\text{green or yellow}) = 0.25 + 0.3 = 0.55$

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Worked Example 2: Jumper Selection Problem

Susie has different coloured jumpers and selects one at random. Here's the probability table:

Table

Part (a): Work out the value of x

All probabilities must sum to 1: $P(\text{Red}) + P(\text{Black}) + P(\text{Blue}) + P(\text{Pink}) = 1$ $0.15 + 0.25 + 0.32 + x = 1$ $0.72 + x = 1$ $x = 1 - 0.72 = 0.28$

Part (b): Work out the probability that she selects a red jumper or a black jumper

Since selecting red and selecting black are mutually exclusive: $P(\text{red or black}) = P(\text{red}) + P(\text{black})$ $P(\text{red or black}) = 0.15 + 0.25 = 0.40$

Key exam tips

Understanding probability requires careful attention to the language and logic involved. Here are the most important strategies to master:

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Essential Exam Strategies:

Check your work: Always verify that all probabilities in a complete set add up to 1.
Identify mutually exclusive events: Ask yourself "Can these events happen at the same time?" If no, they're mutually exclusive.
Use the complement: Sometimes it's easier to calculate P(not A) = 1 - P(A) rather than finding P(A) directly.
Watch out for the word "or": In probability, "or" usually means addition (for mutually exclusive events).
Common trap: Don't add probabilities for events that can occur together - this only works for mutually exclusive events.

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Key Points to Remember:

Mutually exclusive events cannot occur at the same time
For mutually exclusive events: $P(A \text{ or } B) = P(A) + P(B)$
All probabilities in a complete system must add up to 1
$P(\text{not A}) = 1 - P(A)$ is a powerful tool for finding complements
Always check that your calculated probabilities make sense and fall between 0 and 1

Mutually exclusive and exhaustive events (AQA GCSE Statistics): Revision Notes