Probability

Strand 1 — Probability

  • An outcome is a result of an experiment, e.g. rolling a 4 on a dice.
  • The sample space for a fair dice is \( \{1,2,3,4,5,6\} \).
  • Fundamental Principle of Counting: if one event has \(m\) outcomes and another has \(n\), together they have \(m\times n\) outcomes.
  • The probability scale goes from 0 (impossible) to 1 (certain).
  • For equally likely outcomes: \[ P(\text{event})=\dfrac{\text{number of favourable outcomes}} {\text{number of possible outcomes}}. \]

Fundamental Principle of Counting

First Year • Probability & Counting

When we make a sequence of choices, we count all possible outcomes by multiplying the number of options at each step.

1

Step 1 – First choice

Choose a drink. There are 3 options: water, juice, milk.

2

Step 2 – Second choice

Then choose a snack. There are 4 options: fruit, crisps, biscuit, yoghurt.

3

Total combinations

Each drink pairs with each snack → 3 × 4 = 12 different combos.

Rule: If one choice can be made in m ways and another in n ways, then there are m × n possible outcomes.

Quick Check

Multiple-choice quiz
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The Probability Scale

First Year • Probability

The probability scale goes from 0 (impossible) to 1 (certain). Probabilities can be written as fractions, decimals, or percentages between 0 and 1.

0 → Impossible (cannot happen).
½ → Even chance (50–50).
1 → Certain (will happen).
Probability scale from 0 to 1
0 0.25 0.5 0.75 1
Impossible Unlikely Even chance Likely Certain
Example: the probability of rolling an even number on a fair dice is 3 out of 6. That is ½ = 0.5, which is an even chance.

Quick Check

Where does it go on the scale?
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The Probability of an Event Happening

First Year • Probability

Probability tells us how likely an event is to happen. We write the probability of an event as a number between 0 (impossible) and 1 (certain).

Outcome
A single result of an experiment.
Example: rolling a 4 on a dice.
Sample space
The set of all possible outcomes.
Example: for a fair dice the sample space is {1, 2, 3, 4, 5, 6}.
Event
A set of outcomes we are interested in.
Example: “rolling an even number” = {2, 4, 6}.
Probability of an event
Probability = number of favourable outcomes ÷ number of possible outcomes. The answer is always between 0 and 1.
Example: A fair dice is rolled. There are 6 possible outcomes in total. The event “rolling a 4” has 1 favourable outcome (4). So the probability is 1 ÷ 6, written as 1/6.

Quick Check

Probability of an event
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Probability with Coloured Counters

Question 1 of 5 Score: 0 / 5

Seven counters (blue, red and yellow) are in a bag. One counter is chosen at random.

Current counters:
Probabilities are based on the seven counters shown above. © onlinemaths.org

Tossing Two Coins

Sample space • HH, HT, TH, TT

What are the possible outcomes when two coins are tossed?

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Heads are shown in green, tails in red. Try several tosses and list the outcomes you see.

Sample Spaces – Tossing Two Coins

First Year • Probability

A sample space lists all the possible outcomes of an experiment. For probability, this helps us see every outcome clearly and count them.

Outcome
A single result of an experiment.
Example: “Head then Tail”.
Sample space
The set of all possible outcomes.
Example: for one coin, {H, T}.
Event
A collection of outcomes that we are interested in.
Example: “at least one head” when tossing two coins.
Example: Tossing two coins. Each coin can land on Head (H) or Tail (T). The sample space is:
{HH, HT, TH, TT}. There are 4 possible outcomes in total.
To find the probability of an event, we count how many outcomes in the sample space make the event happen (the favourable outcomes) and divide by the total number of outcomes.

Interactive: Toss Two Coins

Build the sample space

Click the buttons to toss two fair coins. Watch how the four outcomes HH, HT, TH, TT appear and how the experimental probabilities move towards the theoretical value of 1/4 for each outcome.

Sample space: { HH, HT, TH, TT }
Total tosses: 0
Last outcome: –

Outcome counts and experimental probabilities

HH0
Experimental probability: 0
HT0
Experimental probability: 0
TH0
Experimental probability: 0
TT0
Experimental probability: 0

Quick Check

Sample spaces with two coins
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Rolling Two Dice

Sample space

What are the possible outcomes when two dice are rolled? Use the simulator to explore the combinations and notice how they form 36 outcomes.

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Each die shows a number between 1 and 6. Try several rolls to see all possible pairs.

Video Tutorials