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Math · Statistics & Probability

Basic probability

Calculate the likelihood of an event from a sample space, including independent and mutually exclusive events — probability is really just counting, dressed up as a fraction. Every question comes with a written walkthrough of exactly what was counted.

Practise basic probability nowSee sample questions
3
Difficulty tiers
35s
Target pace
56%
Average first-attempt score

What's covered

  • Basic probability — divide the number of favorable outcomes by the total number of possible outcomes in the sample space.
  • Independent events — multiply the individual probabilities when one event's outcome doesn't affect the other's, like separate coin flips.
  • Dependent events — adjust the second event's probability to reflect that the sample space changed after the first event occurred, like drawing cards without replacement.
  • Mutually exclusive events — add the individual probabilities when two events can't both happen at once, since there's no overlap to subtract.
  • The complement rule — find the probability an event doesn't happen by subtracting the probability it does happen from 1 — often faster than counting the event directly.

Where students lose marks

Multiplying dependent events like independent ones

Once an item is removed from a sample space and not replaced, the total for the next draw is smaller — reusing the original total treats a dependent event as if it were independent.

Adding probabilities for events that aren't mutually exclusive

If two events can happen at the same time, adding their probabilities double-counts the overlap — that case needs P(A)+P(B)−P(A and B) instead.

Forgetting the complement subtracts from 1, not from a count

The complement rule works on probabilities, which sum to 1 — mixing it up with a count of outcomes produces a number that isn't a valid probability at all.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a blue marble?

Sample 02Core

A standard deck of 52 cards has 13 cards of each suit. Two cards are drawn without replacement. What is the probability both cards are hearts?

Sample 03Advanced

A jar contains 4 red, 3 blue, and 5 green balls. Two balls are drawn without replacement. What is the probability that the two balls are NOT the same color?

How to practise this

Probability problems are counting problems wearing a fraction — the fastest gains come from correctly identifying the sample space before any arithmetic starts.

Identify independent, dependent, or mutually exclusive before calculating

Name which category a problem falls into before choosing whether to multiply, adjust the denominator, or add. Untimed practice is where that classification gets fast.

Then 35 seconds a question

Move to timed sessions once classification is automatic. Multi-step probability problems run longer since there's often more than one event to track.

Fold into a mock

Pair basic probability with counting problems in a mock — many probability questions need a combination or permutation just to find the sample space size.

Stop guessing the category.
Start naming it first.

  • A sample-space walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your basic probability accuracy and pace over time
Start practising free

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Related Math subtopics

Counting problems (combinations and permutations)Mean, Median, ModeAll of Math