The AND and OR Rules

Grade 5

The AND rule — both events happen

When you want the probability of event A and event B both happening, and the two events are independent (one doesn't affect the other), you multiply:

P(A and B) = P(A) × P(B)

Independent means the result of A has no effect on the probability of B — like flipping a coin twice, or rolling two separate dice.

The OR rule — at least one event happens

When you want the probability of A or B happening, and the events are mutually exclusive (they can't both happen at the same time), you add:

P(A or B) = P(A) + P(B)

Mutually exclusive means if one happens, the other can't — like rolling a 3 or a 5 on a single dice.

Worked example

EXAMPLE
A fair dice is rolled and a fair coin is flipped. Find (a) P(6 and heads), (b) P(odd number or tails).
(a) These are independent events. P(6) = 1/6, P(heads) = 1/2. P(6 and heads) = 1/6 × 1/2 = 1/12. (b) "Odd number" and "tails" can both happen on the same roll — they're not mutually exclusive. So the simple OR rule doesn't apply here; use a sample space or tree diagram to list paths: (odd, H), (odd, T), (even, T). That's better handled case by case rather than just adding. This is a reminder that the simple addition rule only works when events can't both occur at once.

Check: are they really mutually exclusive?

The most common mistake is adding probabilities when the events can both happen. If events overlap, the simple OR rule overcounts. Check whether both events can occur at once before you decide to add or multiply.
Nova
Try this one

A bag contains 3 red counters and 7 blue counters. A counter is taken out at random, its colour is recorded, and it is put back in the bag. A second counter is then taken out at random.

(a) Complete the tree diagram below by filling in the four missing probabilities.

First pick           Second pick

Red (\( \frac{3}{10} \)) ——— Red ( __ )
Red ( __ ) <
Red (\( \frac{3}{10} \)) ——— Blue ( __ )

Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Red ( __ )
Blue ( __ ) <
Blue (\( \frac{7}{10} \)) ——— Blue ( __ )

[2]

(b) Work out the probability that both counters are the same colour. [3]

Nova's hint:

This uses tree diagrams with replacement — because the counter goes back, each pick is independent and the probabilities stay the same on every branch.

Ask yourself: 'How do I combine the probabilities along a branch to find the probability of two events both happening?'

For example, if P(A) = 1/4 and P(B) = 1/4 along a branch, the probability of that route is 1/4 × 1/4.

Try to Solve it with Nova?