Leaving Cert Probability (Higher Level): Binomial Distribution & Conditional Probability

Probability covers counting principles, conditional probability, expected value and the binomial distribution. It is examined on Paper 2 and links to statistics.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

Probabilities always lie between 0 and 1.
Conditional probability measures the chance of an event given another has occurred.
Expected value is the long-run average outcome of a random experiment.
Probability is examined on Paper 2.

Probability explained

The Fundamental Principle of Counting

If one task can be done in $m$ ways and another task can be done in $n$ ways, then both tasks together can be done in $m \times n$ ways.

Example: If you have 3 shirts and 4 pants, you can make $3 \times 4 = 12$ different outfits.

This principle extends to multiple tasks: $n_1 \times n_2 \times n_3 \times \ldots$

Factorials

The factorial of a positive integer $n$ is the product of all positive integers up to $n$ : $n! = n \times (n-1) \times (n-2) \times \ldots \times 3 \times 2 \times 1$

Examples: - $3! = 3 \times 2 \times 1 = 6$ - $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$ - $0! = 1$ (by definition)

Factorials grow very quickly!

Permutations: Arranging Objects

A permutation is an arrangement of objects where order matters.

The number of ways to arrange $n$ distinct objects is $n!$

Example: How many ways can 4 books be arranged on a shelf? $4! = 4 \times 3 \times 2 \times 1 = 24 \text{ ways}$

Arranging $r$ objects from $n$ : $^nP_r = \frac{n!}{(n-r)!}$

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Key formulas

Name	Formula	Description
Basic Probability	$P(A) = \frac{n(A)}{n(S)}$	Probability equals favorable outcomes over total outcomes
Complement Rule	$P(A') = 1 - P(A)$	Probability of event not occurring
Addition Rule	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Probability of A or B occurring
Mutually Exclusive	$P(A \cup B) = P(A) + P(B)$	When events cannot occur together
Multiplication Rule	$P(A \cap B) = P(A) \times P(B\|A)$	Probability of A and B occurring
Independent Events	$P(A \cap B) = P(A) \times P(B)$	When occurrence of one does not affect other

Worked examples

Worked example 1

What is the probability of rolling a 6 on a fair die?

There is 1 favorable outcome (rolling a 6) out of 6 possible outcomes.

Answer:

1/6

Worked example 2

Calculate $6!$

Step by step: $6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720$

Answer:

720

Where students lose marks

Adding probabilities that should be multiplied (and vice versa).
Forgetting to account for 'without replacement' in tree diagrams.
Misreading 'at least one' questions (use the complement).

Frequently asked questions

What is conditional probability?

Conditional probability is the probability of an event occurring given that another event has already happened, written P(A given B).

What is expected value?

Expected value is the average result you would expect over many repetitions, found by multiplying each outcome by its probability and adding them up.

Is probability on Paper 1 or Paper 2?

Probability is examined on Paper 2.

Authoritative sources

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