Edexcel International A Level
Statistics 1 (YMA01)
Further Probability
Advanced revision guide covering conditional probability, set notation, Venn diagrams, tree diagrams, two-way tables, and probability formulae
Set Notation in Probability
Set Notation: A formal way of writing groups of mathematical entities (like numbers or events) that share a common feature.
✓ Common Symbols:
- S, U, ξ, ℰ: Universal set (sample space in probability)
- A, B, C: Individual events
- A': Complement of A ("not A")
- A ∩ B: Intersection ("A and B")
- A ∪ B: Union ("A or B")
- ∅: Empty set (no elements)
Basic Probability Rules with Set Notation
P(A ∩ B) = P(A) × P(B) ... for independent events
P(A ∪ B) = P(A) + P(B) ... for mutually exclusive events
P(A') = 1 − P(A) ... complement rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B) ... addition formula
A ∩ B (Intersection - "A and B")
A' (Complement - "not A")
What is Conditional Probability?
Conditional Probability: The probability of an event happening given that another event has already occurred. Denoted by P(A|B), read as "probability of A given B".
✓ Key Concept:
When event B has already occurred, the sample space is reduced to just the outcomes of event B. We then find the probability of A within this reduced sample space.
P(A|B) = P(A ∩ B) / P(B)
This rearranges to:
P(A ∩ B) = P(B) × P(A|B)
💡 For Independent Events: If A and B are independent, then P(A|B) = P(A). The fact that B happened doesn't affect the probability of A.
Finding Conditional Probabilities from Venn Diagrams
Key Strategy
- For P(A|B), first identify event B (the "given" event)
- Shade event B — this becomes your new sample space
- Within the shaded B region, shade event A ∩ B (double shaded area)
- P(A|B) = "double shaded area" / "all B shading"
Finding P(A|B) Using Venn Diagrams
Worked Example Strategy
Using Mini-Venn Diagrams
Tip: When finding P(A|B), P(B|A'), or other conditional probabilities:
- Draw a small "mini-Venn" diagram for each part
- Clearly shade the "given" event first
- Then shade the event you want within that region
- Use different colors or shading styles to show "double shading"
Understanding Tree Diagrams with Conditional Probability
✓ Key Points:
- P(B|A) appears on the branch after event A
- These are the conditional probabilities given that A occurred
- Different branches may have different conditional probabilities
- All probabilities from a point must sum to 1
Tree Diagram Structure with Conditional Probability