Independent events Revision Notes for Edexcel GCSE Maths

Independent events

What are independent events?

Independent events are two or more events where the outcome of one event does not affect the outcome of the other event. This means that what happens in the first event has no influence on what happens in the second event.

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The key characteristic of independent events is that knowledge of one outcome provides no information about the other outcome. Each event maintains its original probability regardless of what happens with other events.

For example, if you flip a coin twice, the result of the first flip does not change the probability of getting heads or tails on the second flip. Each flip is independent of the other.

Calculating probability for independent events

When dealing with independent events, you can find the probability of both events happening by multiplying their individual probabilities together.

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

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This multiplication rule only works when the events are truly independent of each other. If there's any connection between the events, this formula cannot be used.

Worked example: spinner problem

Let's look at a spinner with different coloured sections:

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Worked Example: Spinner Probability

Problem: Huan spins a spinner twice. What is the probability that he:

(a) Gets blue then green
(b) Gets blue on both spins

Solution:

(a) $P(\text{blue then green}) = P(\text{blue}) \times P(\text{green}) = \frac{7}{8} \times \frac{1}{8} = \frac{7}{64}$
(b) $P(\text{blue then blue}) = P(\text{blue}) \times P(\text{blue}) = \frac{7}{8} \times \frac{7}{8} = \frac{49}{64}$

Key insight: Each spin is independent - the first result doesn't affect the second spin's probabilities.

Using tree diagrams for independent events

Tree diagrams are a visual way to show independent events and calculate their combined probabilities. They provide a systematic approach to organising and solving complex probability problems involving multiple events.

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Rules for tree diagrams:

Write the probabilities on each branch
For each pair of branches from the same point, the probabilities must add up to 1
To find the probability of a specific outcome, multiply along the branches
To find the probability of multiple possible outcomes, add up the individual probabilities

Tree diagram structure:

First event -----> Second event
    |                 |
  Branch 1 -----> Outcome A
    |           \---> Outcome B  
  Branch 2 -----> Outcome A
              \---> Outcome B

Worked example: basketball shots

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Worked Example: Basketball Shooting

Problem: Aidan and Chloe each take a basketball shot. Aidan's probability of scoring is 0.3, and Chloe's probability is 0.4. Find the probability that exactly one player scores.

Solution using tree diagram:

Step 1: Identify the two successful outcomes for "exactly one scores"

Aidan scores AND Chloe misses
Aidan misses AND Chloe scores

Step 2: Calculate each probability

$P(\text{Aidan scores, Chloe misses}) = 0.3 \times 0.6 = 0.18$
$P(\text{Aidan misses, Chloe scores}) = 0.7 \times 0.4 = 0.28$

Step 3: Add the probabilities

$P(\text{exactly one scores}) = 0.18 + 0.28 = 0.46$

This shows there are two successful outcomes for "exactly one player scores", so we multiply along each relevant branch then add the results together.

Key steps for exam questions

When approaching independent events problems, follow this systematic approach:

Identify if events are independent (one doesn't affect the other)
Draw a tree diagram if dealing with multiple outcomes
Multiply probabilities along branches for combined events
Add probabilities when you want "this OR that" to happen
Check that probabilities from each branch point add to 1

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Always verify that your events are truly independent before applying the multiplication rule. If one event influences another, you'll need different probability techniques.

Summary

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Key Points to Remember:

Independent events means one outcome doesn't affect the other
Multiply probabilities to find the chance of both events happening: $P(A \text{ and } B) = P(A) \times P(B)$
Tree diagrams help visualise independent events clearly
Branch probabilities from the same point always add to 1
Multiply along branches, add between different paths for calculations

Independent events (Edexcel GCSE Maths): Revision Notes