Combined events and conditional branches

“Cannot both happen” differs from “do not affect each other”

Two events are mutually exclusive when they cannot occur together. On one fair die roll, “1” and “6” are mutually exclusive, so the probability of either is 1/6 + 1/6 = 1/3.

Events are independent when learning one occurred does not change the other’s probability. On two independent fair die rolls, “first roll is 1” and “second roll is 6” are independent and can both happen, with probability 1/6 × 1/6 = 1/36.

Two mutually exclusive events with positive probabilities are not independent: knowing one happened makes the other impossible. Do not use these words interchangeably.

Add disjoint routes to the required outcome

A bag contains three red and two blue counters. Two are drawn without replacement. A probability tree changes its second-stage denominators because only four counters remain.

Ordered route	Probability
Red then red	(3/5)(2/4) = 3/10
Red then blue	(3/5)(2/4) = 3/10
Blue then red	(2/5)(3/4) = 3/10
Blue then blue	(2/5)(1/4) = 1/10

Exactly one red occurs on two disjoint routes, RB and BR. Add them to obtain 3/5. Check all four route probabilities total 1. With replacement, the second-stage probabilities stay 3/5 and 2/5 instead: that is a different experiment.

Overlapping events need the overlap removed once

In general P(A or B) = P(A) + P(B)-P(A and B). For one die, even outcomes are 6 and outcomes above 3 are 6. Their union is 6, probability 4/6. Adding 3/6 and 3/6 without subtracting the shared 2/6 counts 4 and 6 twice.

Check your understanding 1 The second draw depends on the first

A bag holds four white and two black counters. Two are drawn without replacement. Find the probability that both are black and the probability that they have different colours.

Your answer or drawing on paper

Show answer

Both black: (2/6)(1/5) = 1/15. Different colours: (4/6)(2/5) + (2/6)(4/5) = 8/15. The second draw has five counters, and its colour counts depend on the first draw.

Use a possibility table when it is clearer

For two fair dice, a 6-by-6 table lists 36 equally likely ordered pairs. There are five pairs summing to 6, so probability 5/36. A tree and a table describe the same sample space; choose the representation that exposes the required routes without omissions.