Skip to content
Marlbridge

AP® Statistics · Unit 2 · Topic 2.4

Introduction to Probability: Revision Notes (Statistics 2.4)

One-page recap of sample spaces, equally likely outcomes, the 0-to-1 rule, P(S) = 1 and the complement rule, with the mistakes that cost marks.

Short on time? This page is the recap. For explanations and worked examples, use the full study guide.

Recap

  • The sample space S is the set of all possible outcomes, with no overlaps. One outcome happens on every trial.
  • An event E is a set of outcomes. Its probability is written P(E).
  • With equally likely outcomes, find P(E) by counting.
  • The complement of E (written Eᶜ, E′ or Ē) is “E does not happen”.
  • For two-stage processes, list outcomes as pairs in a grid or a systematic list.

Key relationships

Rule Statement Use it to…
Sample space P(S) = 1; probabilities of all separate outcomes add to 1 find a missing probability
Range 0 ≤ P(E) ≤ 1 (0 = impossible, 1 = certain) check every answer
Equally likely outcomes P(E) = outcomes in E ÷ outcomes in S calculate an exact probability
Random choice from a table P(category) = count ÷ total turn relative frequencies into probabilities
Complement P(Eᶜ) = 1 − P(E) handle “not” and “at least one”
At least one P(at least one) = 1 − P(none) avoid counting many outcomes

Assumptions

  • The counting formula needs equally likely outcomes: a fair die or coin, equal sectors, or one individual chosen at random.
  • Outcomes in the sample space must not overlap, and none may be missing.
  • Order matters when the stages are different (day 1 then day 2; die then spinner).

Mistakes to avoid

  1. “Three outcomes, so 1/3 each” when the outcomes are not equally likely.
  2. Counting totals or scores instead of the equally likely pairs behind them.
  3. Missing or double-counting outcomes; use a grid.
  4. Answers below 0 or above 1. Always impossible.
  5. Wrong complement: the complement of “at least 2” is “at most 1”.
  6. Writing a percentage as a probability (87.5 instead of 0.875).

Quick self-check

  1. P(E) = 0.27. Find P(Eᶜ). (1 − 0.27 = 0.73)
  2. A fair spinner has 5 equal sectors numbered 1 to 5. Find P(even number). (2 of 5 outcomes: 2/5 = 0.4)
  3. What is the complement of “at least one of the three days is sunny”? (None of the three days is sunny.)
  4. Two fair coins are tossed. Find P(at least one head). (S = {HH, HT, TH, TT}, four equally likely outcomes. P(no heads) = 1/4, so P(at least one head) = 1 − 1/4 = 3/4 = 0.75.)
  5. A model gives P(red) = 0.45, P(blue) = 0.35 and P(green) = 0.3 for the only three outcomes. What is wrong? (The probabilities add to 1.1, not 1.)

Next: practice questions.

AP® and Advanced Placement® are trademarks registered by the College Board, which is not affiliated with, and does not endorse, this website. Marlbridge is an independent tuition provider. It is not an authorized AP course provider, an AP school or an exam centre, and these resources are not official College Board materials. Reading them or doing simulations does not meet the laboratory requirement of an AP science course.