Conditional Probability Revision Notes for AQA GCSE Maths

Conditional probability

What is conditional probability?

Conditional probability deals with situations where one event influences another - these are called dependent events. When we talk about conditional probability, we're asking: "What's the chance of event A happening, knowing that event B has already occurred?"

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The conditional probability of event A given that event B has happened is written as $P(A|B)$ or "P(A given B)". This tells us how likely A is to occur when we know B has already taken place.

Recognising conditional probability problems

There are several clues that indicate you're dealing with a conditional probability question:

Watch out for "without replacement" - This is a major hint that events are dependent on each other. When items are removed and not put back, it changes the probabilities for subsequent events.

Look for connected events - Questions often involve scenarios where the outcome of one event affects what happens next, such as drawing cards from a deck or selecting items from a bag.

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Independent vs dependent events - If events A and B are independent (one doesn't affect the other), then $P(A|B) = P(A)$ and $P(B|A) = P(B)$ . This means knowing B happened doesn't change the probability of A occurring.

The AND rule for conditional probabilities

When you need to find the probability of both events A and B happening together, you can use the AND rule:

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

This formula tells us that the probability of both events occurring equals the probability of the first event multiplied by the probability of the second event happening given that the first event has already occurred.

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Worked Example: Reading and Tiredness

Let's say someone either watches TV or reads before bed. The probability they watch TV is 0.3. If they read, the probability they won't be tired the next day is 0.8. What's the probability they both read and aren't tired the next day?

Step 1: Identify the events

Event A: "reads"
Event B: "isn't tired"

Step 2: Work out the probabilities

$P(A) = P(\text{reads}) = 1 - 0.3 = 0.7$
$P(B|A) = P(\text{isn't tired given reads}) = 0.8$

Step 3: Apply the AND rule

$P(A \text{ and } B) = P(A) \times P(B|A) = 0.7 \times 0.8 = 0.56$

Using tree diagrams for conditional probability

Tree diagrams are extremely helpful for visualising conditional probability problems. The key insight is that probabilities on branches change depending on what happened in previous events.

Tree diagram method

When drawing a tree diagram for conditional probability:

Draw the first set of branches showing all possible outcomes for the first event
For each branch, draw the second set but remember that the probabilities for the second event depend on what happened in the first event
Multiply along branches to find the probability of specific combinations
Add probabilities when you want to find the chance of different ways the same outcome can occur

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Worked Example: Coloured Discs

Imagine a box with 5 red discs and 3 green discs. Two discs are taken without replacement. The tree diagram would show:

First pick: $\frac{5}{8}$ chance of red, $\frac{3}{8}$ chance of green

Second pick depends on first pick:

If first was red: $\frac{4}{7}$ chance of red, $\frac{3}{7}$ chance of green
If first was green: $\frac{5}{7}$ chance of red, $\frac{2}{7}$ chance of green

To find P(both discs same colour):

$P(\text{both red}) = \frac{5}{8} \times \frac{4}{7} = \frac{20}{56}$
$P(\text{both green}) = \frac{3}{8} \times \frac{2}{7} = \frac{6}{56}$
$P(\text{both same colour}) = \frac{20}{56} + \frac{6}{56} = \frac{26}{56} = \frac{13}{28}$

Problem-solving strategies

When tackling conditional probability questions, follow these essential steps:

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Step-by-step approach:

Identify whether events are dependent - Look for clues like "without replacement"
Define your events clearly - Label what A and B represent
Consider drawing a tree diagram - This helps visualise the problem, especially for complex scenarios
Use the AND rule when finding probabilities of multiple events occurring together
Remember to add probabilities when there are multiple ways to achieve the same outcome

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Key Points to Remember:

Conditional probability asks "what's the chance of A happening, knowing B has occurred?"
"Without replacement" is a key indicator of conditional probability problems
For independent events: $P(A|B) = P(A)$
The AND rule: $P(A \text{ and } B) = P(A) \times P(B|A)$
Tree diagrams are invaluable for visualising conditional probability scenarios
Always clearly define your events before starting calculations

Conditional Probability (AQA GCSE Maths): Revision Notes