Leaving Cert Maths Probability: Rules and Worked Examples
Quick answer: Leaving Cert probability starts with counting (arrangements and choices), then the addition rule, P(A or B) = P(A) + P(B) − P(A and B), and the multiplication rule for independent events. Higher Level adds Bernoulli trials and expected value. Most marks are lost on setting up the problem, not on the arithmetic.
Counting
Fundamental principle: 3 shirts and 4 pairs of trousers give 3 × 4 = 12 outfits.
Arrangements (order matters): 5 people in a row can be arranged in 5! = 120 ways. Picking and ordering 3 from 7: 7 × 6 × 5 = 210.
Choices (order doesn't matter): choosing 3 from 7 is 7C3 = 210 ÷ 3! = 35.
Ask: does the order change the outcome? If yes, arrange. If no, choose.
The addition rule
P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B).
0.5 + 0.4 − 0.2 = 0.7.
If A and B are mutually exclusive, P(A and B) = 0, so you just add.
Independent events
A and B are independent when P(A and B) = P(A) × P(B).
A coin is tossed and a die is rolled. Find P(head and a six).
½ × ⅙ = 1/12.
Bernoulli trials (Higher Level)
A fair die is rolled 4 times. Find the probability of exactly 2 sixes.
There are 4C2 = 6 ways to place the two sixes.
P = 6 × (1/6)² × (5/6)² = 150/1296 ≈ 0.116.
Expected value
A game costs nothing to play. You win €10 with probability 0.2, and lose €3 with probability 0.8. What's the expected value?
E = 10(0.2) + (−3)(0.8) = 2 − 2.4 = −€0.40.
On average you lose 40 cent a game, so it's not a fair game.
Exam tips
Write the event in words before writing numbers.
Check the formulae and tables booklet for which formulas are given.
Sense-check: every probability is between 0 and 1.
Tips are MMS editorial guidance.
Common mistakes
Using arrangements when order doesn't matter.
Adding probabilities that overlap without subtracting P(A and B).
Multiplying events that aren't independent.
When extra support might help
Support is worth considering when probability questions are hard to start, or before the mocks.
A grind can't guarantee a grade, but can build method and confidence. Math Make Smart offers online maths grinds and Leaving Cert support, and you can book a free grind. See also our Paper 1 vs Paper 2 guide.
Frequently asked questions
What is the addition rule?
P(A or B) = P(A) + P(B) − P(A and B).
When are events independent?
When P(A and B) = P(A) × P(B).
What is expected value?
The average result per trial over many trials.