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Probability — What Are the Chances?

Learn probability from scratch — from coin flips and dice rolls to tree diagrams and the AND/OR rules. With everyday examples and worked problems.

Probability is about measuring how likely something is to happen. It's a number between 0 (impossible — like rolling a 7 on a normal dice) and 1 (certain — like the sun rising tomorrow).

The Basic Rule

Probability = Number of favourable outcomes ÷ Total number of outcomes

Simple Examples

EventFavourableTotalProbability
Flipping heads121/2
Rolling a 6161/6
Rolling an even number361/2
Drawing a heart from a deck13521/4

The AND Rule — Both Things Happen

If two events are independent (one doesn't affect the other), multiply their probabilities.

Worked Example 1 — Coin and Dice

What's the probability of flipping heads AND rolling a 6?

  1. P(heads) = 1/2
  2. P(six) = 1/6
  3. P(heads AND six) = 1/2 × 1/6 = 1/12

The OR Rule — Either Thing Happens

If the events can't happen at the same time (mutually exclusive), add their probabilities.

Worked Example 2 — Rolling a Dice

What's the probability of rolling a 2 OR a 5?

  1. P(2) = 1/6
  2. P(5) = 1/6
  3. P(2 OR 5) = 1/6 + 1/6 = 2/6 = 1/3

Tree Diagrams

Tree diagrams are brilliant for keeping track of multiple events. Here's how they work:

  1. Draw a branch for each outcome of the first event
  2. From the end of each branch, draw branches for the second event
  3. Write the probability on each branch
  4. Multiply along the branches to find the probability of that path
  5. Add paths together if you want "this OR that"

Worked Example 3 — Two Coin Flips

What's the probability of getting exactly one head in two flips?

Paths: HH, HT, TH, TT (each with probability 1/4)

  1. Paths with exactly one H: HT and TH
  2. P = 1/4 + 1/4 = 1/2

Expected Frequency

If you roll a dice 300 times, how many sixes would you expect? Expected frequency = probability × number of trials = 1/6 × 300 = 50.

Try It Yourself

  1. A bag has 3 red and 7 blue balls. What's the probability of picking red? (Answer: 3/10)
  2. What's the probability of flipping 3 heads in a row? (Answer: 1/8)
  3. You roll a dice 180 times. How many times do you expect an even number? (Answer: 90)

Calculate probabilities for more complex scenarios.

→ Open the Probability Calculator