AP Statistics · Unit 2 · Topic 2.4

Introduction to Probability Study Lab

When all outcomes are equally likely, the probability of an event is the number of outcomes in the event divided by the number in the sample space. Pick an experiment, choose an event, and see every outcome that counts. Then build a probability model and check that it follows the rules.

1. Count outcomes in a sample space

Choose an experiment and an event. Highlighted outcomes are in the event.

P(E) = outcomes in E ÷ outcomes in the sample space P(not E) = 1 − P(E)
Experiment
Event

Sample space

Show your work

Outcomes in sample space
Outcomes in event
P(event)
P(complement)

Build a probability model

A valid model gives every outcome a probability between 0 and 1, and the probabilities add to exactly 1. Edit the values, leave one blank to solve for it, and check the outcomes in your event.

Preset models

Check the model

2. Key ideas

Sample space

Every possible outcome

The sample space S is the list of all possible outcomes of a random process. Listing it carefully, often in a grid or tree, is the first step in most probability problems.

Event

A set of outcomes

An event is any collection of outcomes from the sample space, such as "the sum is 7." An event happens if the outcome of the process is one of its outcomes.

Equally likely

Count and divide

Only when all outcomes are equally likely can you find P(E) by counting. Sums of two dice are not equally likely (7 is more likely than 2), but the 36 ordered pairs are.

Range

Between 0 and 1

Every probability is between 0 (impossible) and 1 (certain). The probabilities of all the outcomes in a sample space add to 1.

Complement

"Not E"

The complement Eᶜ is every outcome not in E. P(Eᶜ) = 1 − P(E). It is often easier: P(at least one 6) = 1 − P(no 6s).

Interpreting

Long-run meaning

P(E) = 0.25 means that if the process were repeated many, many times, E would happen in about 25% of the repetitions. It does not promise exactly 1 out of every 4.

3. Worked examples

QuestionSetupWorkAnswer
Roll two fair dice. P(sum is 8)?36 equally likely ordered pairs(2,6), (3,5), (4,4), (5,3), (6,2) → 5 ÷ 360.139
Flip three coins. P(at least one head)?8 outcomes; complement is TTT1 − 1/80.875
Blood types: O 0.45, A 0.40, B 0.11. P(AB)?Probabilities must add to 11 − (0.45 + 0.40 + 0.11)0.04
Draw one card. P(not a heart)?13 hearts out of 521 − 13/520.75
Is P(A) = 0.5, P(B) = 0.3, P(C) = 0.3 a valid model?Check the sum0.5 + 0.3 + 0.3 = 1.1No, the sum is not 1

4. On the FRQ

How to earn the points

  • Define the event in words before calculating: "Let E = the sum of the two dice is at least 10."
  • Show the counting or the complement: "P(at least one defect) = 1 − P(no defects) = 1 − 0.92 = 0.08."
  • Interpret a probability as a long-run relative frequency, in context.
  • When you list a sample space, make it complete and organized (grid, table, or tree) so the grader can count with you.

Common mistakes

  • Treating outcomes as equally likely when they are not, such as saying P(sum = 2) = 1/11 because there are 11 possible sums.
  • Counting (3, 5) and (5, 3) as one outcome. With two dice, order matters for equally likely outcomes.
  • Writing "at least one" as "exactly one."
  • Reporting a probability greater than 1 or less than 0. If you get one, check your work.

5. Practice questions

AP-style multiple choice. Pick an answer to see the explanation.

0 / 0 correct

6. Key terms

The vocabulary for topic 2.4. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 2 terminology guide.