AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/2/2-4)
Unit 2 · Topic 2.4
2.4 Introduction to Probability
This topic sets up the basic rules of probability: the sample space, equally likely outcomes, the 0-to-1 scale and the complement rule. The complement rule in particular turns many "at least one" problems into one quick subtraction.
Key terms
- sample space
- equally likely outcomes
- probability P(E)
- complement
Sample space
The sample space is the list of every possible outcome of a random process, with no overlaps. For one roll of a six-sided die it's {1, 2, 3, 4, 5, 6}. For rolling two dice, it's all 36 ordered pairs, from (1, 1) to (6, 6).
Something always happens, so the probability of the whole sample space is 1. A table or grid is the easiest way to list outcomes for two-stage processes like two dice.
Equally likely outcomes
When every outcome is equally likely, P(E) = (number of outcomes in E) ÷ (total number of outcomes). Rolling two dice, a sum of 7 happens 6 ways: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). So P(sum = 7) = 6/36 = 1/6.
This formula only works for equally likely outcomes. The possible sums 2 through 12 are not equally likely, so P(sum = 7) is not 1/11.
Rules every probability follows
- Every probability is between 0 and 1, inclusive. 0 means impossible; 1 means certain.
- The probabilities of all outcomes in the sample space add to 1.
- Complement rule: P(not E) = 1 − P(E). The complement of E, written Eᶜ or E′, is everything in the sample space that isn't in E.
Event language
Events are sets of outcomes, so set words carry over. "A and B" means both happen, written A ∩ B (the intersection). "A or B" means at least one happens, written A ∪ B (the union); in statistics, "or" includes the case where both happen. "Not A" is the complement. Topics 2.5 to 2.7 give the rules for each.
Theoretical vs. estimated probability
A theoretical probability comes from a model, like assuming a die is fair and counting outcomes. An estimated probability comes from data: the relative frequency of the event in many real or simulated trials (topic 2.3).
They should agree when the model is right and there are lots of trials. If you roll a die 6,000 times and get 1,410 sixes, the estimate 1,410/6,000 = 0.235 is far from 1/6 ≈ 0.167, which suggests the die isn't fair.
When to use the complement
Use the complement whenever "not E" is easier to count than E. The classic case is "at least one": the complement of "at least one six in two rolls" is "no sixes in two rolls."
Counting directly is slow and easy to mess up. With the complement: there are 5 × 5 = 25 outcomes with no six, so P(no six) = 25/36 and P(at least one six) = 1 − 25/36 = 11/36 ≈ 0.306.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Two dice
Roll two fair six-sided dice. Find P(sum ≥ 10) and P(sum < 10).
Show the solutionHide the solution
- Step 1: There are 36 equally likely ordered outcomes.
- Step 2: Sum 10: (4,6), (5,5), (6,4), which is 3 ways. Sum 11: (5,6), (6,5), which is 2 ways. Sum 12: (6,6), which is 1 way. Total: 6.
- Step 3: P(sum ≥ 10) = 6/36 = 1/6 ≈ 0.167.
- Step 4: "Sum < 10" is the complement, so P(sum < 10) = 1 − 1/6 = 5/6 ≈ 0.833.
Answer: P(sum ≥ 10) = 1/6 ≈ 0.167; P(sum < 10) = 5/6 ≈ 0.833.
- Example 2Calculator allowed
Trap: outcomes that aren't equally likely
A spinner has three regions labeled red, blue and green. A student says P(red) = 1/3 because there are three colors. The red region covers 180° of the circle, blue 120° and green 60°. Find the correct probabilities.
Show the solutionHide the solution
- Step 1: The three colors aren't equally likely, so counting colors doesn't work.
- Step 2: Each probability is the region's share of 360°.
- Step 3: Red: 180/360 = 0.5. Blue: 120/360 ≈ 0.333. Green: 60/360 ≈ 0.167.
- Step 4: Check: 0.5 + 0.333 + 0.167 = 1.
Answer: P(red) = 0.5, P(blue) = 1/3, P(green) = 1/6. The 1/3 answer wrongly assumes equally likely outcomes.
Common mistakes
- Using "favorable ÷ total" when outcomes aren't equally likely.
- Treating (1, 6) and (6, 1) as the same outcome when rolling two dice. They're different ordered outcomes.
- Writing a probability greater than 1 or less than 0 without noticing it's impossible.
- Taking the complement of "at least one" as "exactly one" instead of "none."
On the exam
- "At least one" in a question is a strong hint to use 1 − P(none).
- Probability answers should be between 0 and 1. A quick check catches many arithmetic slips.
Connected topics
Videos
Check yourself
3 questions on 2.4 Introduction to Probability. Pick an answer to see if you got it, and why.
Two fair six-sided dice are rolled. What is the probability that the sum is 8?
A survey finds that 28% of the adults in a town have never visited the town's museum. If an adult is chosen at random, what is the probability that the adult has visited the museum at least once?
A spinner has four colored sections. The probability of landing on red is 0.35, on blue is 0.25 and on green is 0.30. What is the probability of landing on the fourth color, yellow?
0 of 3 answered