Independent events Revision Notes for AQA GCSE Statistics

Independent events

What are independent events?

When we talk about independent events in probability, we mean situations where the outcome of one event has absolutely no influence on the outcome of another event. Think of it this way: if knowing what happened in the first event doesn't help you predict what will happen in the second event, then these events are independent.

For example, when you roll a dice and then flip a coin, the number you get on the dice doesn't change the chances of getting heads or tails on the coin. These are independent events because they don't affect each other.

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The key concept here is independence - one event's outcome has zero effect on another event's probability. This is different from dependent events where the first outcome changes the probabilities for subsequent events.

The multiplication law for independent events

This is the key formula you need to remember for independent events:

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If events A and B are independent, then:

$P(A \text{ and } B) = P(A) \times P(B)$

This rule is called the multiplication law for independent events.

It tells us that when we want to find the probability of two independent events both happening, we simply multiply their individual probabilities together.

Worked examples with two events

Let's look at some clear examples to see how this works in practice:

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Worked Example 1: Rolling a dice twice

Imagine you roll a fair dice twice. What's the probability of getting a 4 on the first roll and then a 6 on the second roll?

Since each roll of the dice is independent (the first roll doesn't affect the second), we can use our multiplication rule:

$P(\text{getting a 4}) = \frac{1}{6}$
$P(\text{getting a 6}) = \frac{1}{6}$
$P(\text{4 then 6}) = P(4) \times P(6) = \frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

The outcome of the first roll doesn't change the probabilities for the second roll, which is exactly what makes these events independent.

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Worked Example 2: Spinning a coin twice

Now let's consider spinning a fair coin twice. What's the probability of getting heads both times?

Again, each coin spin is independent of the other:

$P(\text{getting heads}) = 0.5$
$P(\text{heads and heads}) = P(H) \times P(H) = 0.5 \times 0.5 = 0.25$

The result of the first spin has no effect on what happens with the second spin, so the events remain independent.

Multiple independent events

The multiplication law can be extended when you have three or more independent events. The principle remains the same - you multiply all the individual probabilities together.

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For three independent events A, B and C:

$P(A \text{ and } B \text{ and } C) = P(A) \times P(B) \times P(C)$

This pattern continues for as many independent events as you need to work with.

Comprehensive worked example

Let's work through a more complex example step by step:

Clemmie throws a fair dice and spins a fair coin.

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Worked Example: Part (a) - Probability of getting a 4 and a head

To find $P(\text{4 and head})$ , we multiply the individual probabilities:

$P(\text{getting a 4 on the dice}) = \frac{1}{6}$
$P(\text{getting a head on the coin}) = \frac{1}{2}$
$P(\text{4 and head}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$

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Worked Example: Part (b) - Probability of getting an even number and a head

First, we identify what counts as an even number on a dice: 2, 4, or 6.

$P(\text{getting an even number}) = \frac{3}{6} = \frac{1}{2}$
$P(\text{getting a head}) = \frac{1}{2}$
$P(\text{even number and head}) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

Now Clemmie throws 2 fair dice and spins a fair coin.

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Worked Example: Part (c) - Probability of getting 2 odd numbers and a tail

For each dice, the odd numbers are 1, 3, and 5:

$P(\text{getting an odd number on first dice}) = \frac{3}{6} = \frac{1}{2}$
$P(\text{getting an odd number on second dice}) = \frac{3}{6} = \frac{1}{2}$
$P(\text{getting a tail}) = \frac{1}{2}$
$P(\text{2 odd numbers and a tail}) = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{8}$

Notice how we multiply three probabilities because we have three independent events happening.

Key exam tips

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Important Exam Strategy Points:

Check for independence: Make sure the events truly don't affect each other before using the multiplication rule
Multiply, don't add: For independent events occurring together, always multiply the probabilities
Work step by step: Break down complex problems into smaller parts and identify each individual probability first
Be careful with fractions: Make sure you simplify your final answers properly
Alternative methods exist: You could also solve some of these problems using sample space diagrams, but multiplication is usually quicker for independent events

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Key Points to Remember:

Independent events are those where one event doesn't affect the probability of another event occurring
For independent events A and B: $P(A \text{ and } B) = P(A) \times P(B)$
This multiplication rule extends to any number of independent events
Always multiply probabilities for independent events happening together - never add them
Common examples include multiple coin flips, dice rolls, or draws with replacement

Independent events (AQA GCSE Statistics): Revision Notes