The formula for conditional probability Revision Notes for Edexcel GCSE Statistics

The formula for conditional probability

What is conditional probability?

Conditional probability helps us find the likelihood of one event happening when we already know that another event has occurred. The notation P(B|A) means "the probability of event B happening, given that event A has already happened."

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Think of it this way: when we know something has already occurred, it changes our perspective on what might happen next. For example, if we know it's raining outside, this affects the probability that someone will carry an umbrella.

The conditional probability formula

The fundamental formula for conditional probability is:

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

This formula tells us that to find the probability of B given A, we need to:

Find the probability that both A and B happen together
Divide this by the probability of A happening on its own

Understanding the formula with Venn diagrams

When we look at a Venn diagram, conditional probability becomes clearer. The formula can also be written as:

$P(B|A) = \frac{\text{number of outcomes in A and B}}{\text{number of outcomes in A}}$

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This makes sense because we're only considering the cases where A has happened (the denominator), and among those cases, we want to know how many also have B happening (the numerator).

Rearranging the formula

We can rearrange the conditional probability formula to find other probabilities:

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

This rearranged version is particularly useful when we know the conditional probability and want to find the probability of both events occurring together.

Independent events and conditional probability

When two events are independent, knowing that one has occurred doesn't change the probability of the other.

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For independent events A and B:

$P(A|B) = P(A)$
$P(B|A) = P(B)$

This means that if A and B are independent, then the fact that A has happened has no effect on the probability of B happening.

Worked example: Tiles problem

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Worked Example: Tiles Problem

The scenario: Kim has 20 tiles in total. 10 tiles have the letter X, and of these X tiles, 6 are red. The remaining 10 tiles have the letter Y, and of these Y tiles, 7 are red. All other tiles are white.

Setting up the events:

Event A: "a white tile is selected"
Event B: "a tile with letter X is selected"

Step 1: Find P(A) We need to count the white tiles:

X tiles that are white: $10 - 6 = 4$ white X tiles
Y tiles that are white: $10 - 7 = 3$ white Y tiles
Total white tiles: $4 + 3 = 7$

Therefore: $P(A) = \frac{7}{20}$

Step 2: Find P(B|A) This asks: "Given that we've selected a white tile, what's the probability it has letter X?"

Among the 7 white tiles, 4 have letter X. Therefore: $P(B|A) = \frac{4}{7}$

Step 3: Find P(A and B)
Using our rearranged formula: $P(A \text{ and } B) = P(B|A) \times P(A)$

$P(A \text{ and } B) = \frac{4}{7} \times \frac{7}{20} = \frac{4}{20} = \frac{1}{5}$

Key formula relationships

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Essential Formulas to Remember:

Basic conditional probability: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$
Rearranged form: $P(A \text{ and } B) = P(B|A) \times P(A)$
For independent events: $P(A|B) = P(A)$ and $P(B|A) = P(B)$

Common exam tips

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Common Mistakes to Avoid:

Always check if events are independent - this simplifies calculations significantly
Make sure you identify which event is the "given" condition (this goes after the vertical line |)
When using Venn diagrams, count carefully and double-check your totals
Remember that $P(A \text{ and } B)$ is different from $P(A|B)$ - don't confuse these!
Write out the formula before substituting values to avoid errors

Key takeaways

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Key Points to Remember:

Conditional probability $P(B|A)$ means "probability of B given that A has happened"
The formula is $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$
For independent events, $P(B|A) = P(B)$ because knowing A doesn't affect B
You can rearrange the formula to find $P(A \text{ and } B) = P(B|A) \times P(A)$
Always identify which event is the condition and which is the outcome you're finding

The formula for conditional probability (Edexcel GCSE Statistics): Revision Notes