Probability

Revise Probability for Probability & Statistics 1 S1 (AS Level) — revision notes and instant AI marking.

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Cambridge International AS & A Level Mathematics 9709 — Probability & Statistics 1

Probability

The big idea: Probability is counting with weights: multiply along a path, add across paths, and when you are told something has happened, shrink the sample space to just that.

Summary — What This Chapter Covers

For equally likely outcomes, P(A) = number of outcomes in A ÷ total number of outcomes.
Tree diagrams: multiply along branches, add the end results you want.
Mutually exclusive: P(A ∩ B) = 0, so P(A ∪ B) = P(A) + P(B).
Independent: P(A ∩ B) = P(A) × P(B).
Conditional probability: P(A | B) = P(A ∩ B)/P(B).

1. Basic Probability and Tree Diagrams

Probability

Enumeration and trees

When outcomes are equally likely, list them (a sample-space grid for two dice) or count them with permutations and combinations. P(A′) = 1 − P(A) is often the fastest route to an "at least one" probability.

A tree diagram handles events in sequence. Without replacement, the second-stage probabilities change. Multiply along a path to get the probability of that combination of outcomes; add the paths that satisfy the event.

Worked Example 1

Two fair six-sided dice are thrown. Find the probability that the total is 8.

Worked Example 2

A bag contains 5 red and 3 blue balls. Two are taken at random without replacement. Find the probability that they are the same colour.

Practice Question 1

Three balls are taken without replacement from the same bag. Using combinations, find the probability that all three are red.

2. Exclusive and Independent Events

Exclusive vs independent

Two different ideas

Mutually exclusive events cannot happen together: P(A ∩ B) = 0. Independent events do not affect each other's probabilities: P(A ∩ B) = P(A) × P(B). To test for independence, calculate P(A ∩ B) and compare it with P(A) × P(B) — you must show both numbers. Two events with non-zero probabilities cannot be both exclusive and independent.

Tests Exclusive: P(A ∩ B) = 0 Independent: P(A ∩ B) = P(A)P(B)

Worked Example

Two fair dice are thrown. A is "the first die shows 6" and B is "the total is 7". Are A and B independent? Are they exclusive?

Practice Question 2

P(A) = 0.4, P(B) = 0.5 and P(A ∪ B) = 0.7. Determine whether A and B are independent.

3. Conditional Probability

Conditional