A. Counting and Probability · Probability

Probability: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6About 40 minutes

Learning goals

I can find theoretical, experimental and simulated probabilities, and convert between probability and odds.
I can describe sample spaces and events with set notation, Venn diagrams and tree diagrams.
I can solve problems with complements, mutually exclusive events, independent and dependent events, and conditional probability.

Words to know

sample space
The set of all possible outcomes, S.

complement
A′, the event that A does not happen; P(A′) = 1 − P(A).

mutually exclusive
Events that cannot happen together: P(A ∩ B) = 0.

independent
Events that do not affect each other's probabilities: P(A ∩ B) = P(A) × P(B).

conditional probability
P(B | A) = P(A ∩ B)/P(A), the probability of B given A.

Let's review!

In this unit you learned what a probability measures and how to write it as a fraction, decimal, percent or odds. You described sample spaces (discrete and continuous), calculated theoretical probabilities and probability distributions, and saw experimental probability settle toward theory as trials grow, including with simulations. Then you treated events as sets: complements, intersections and unions, mutually exclusive events, independent and dependent events, and conditional probability, finishing with multi-step problems.

The next unit, Permutations and Combinations, gives you fast ways to count large sample spaces, so you can find probabilities like the chance of a particular poker hand. This review has questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Do two or three sections at a time. Before calculating, name your events and decide which tool fits: a list, a table, a tree, a Venn diagram, or a formula. Keep answers as fractions where possible and round decimals only at the end. For each question you miss, write down why (wrong denominator? forgot the overlap? treated dependent events as independent?) and go back to that lesson. Finish with the 10-question quiz.

How the lessons of this unit fit together. The top rows find probabilities; the bottom rows combine events using sets and rules.

The key rules of the unit on one card. Every one of them can be checked with a small example, such as one die or one card.

Just for fun

Probability of finishing this review: 1, if you start. Odds against finishing it in one sitting: depends entirely on snacks.

Probability as a Measure of Likelihood

Write 7/20 as a decimal and a percent.

Classify: "Out of 500 games, this goalie won 290, so P(win) ≈ 0.58."

P(a part is defective) = 0.006. How many defective parts are expected in 3500?

Sample Spaces: Discrete and Continuous

How many outcomes are there for rolling a die and tossing two coins?

Discrete or continuous: the volume of milk in a carton?

List the sample space for choosing 2 different letters, in order, from {X, Y, Z}.

Theoretical Probability and Probability Distributions

Two dice are rolled. Find P(sum is 5).

One card is drawn. Find P(a black card numbered 2 to 10).

A spinner's sectors are 120°, 120°, 90° and 30°. Find each probability and check the total.

Odds in Favour and Odds Against

Find the odds in favour of drawing a face card.

The odds against an event are 7 : 3. Find its probability.

P(A) = 0.85. Find the odds in favour of A.

Experimental Probability and the Law of Large Numbers

A die was rolled 240 times and showed a 3 on 46 rolls. Find the experimental P(3) and compare with the theoretical value.

Explain why a coin that has landed heads 6 times in a row is not "due" for tails.

Which estimate is more trustworthy: 12 successes in 40 trials, or 300 in 1000?

Designing and Using Simulations

Assign random digits to simulate an event with probability 0.6.

Write a spreadsheet formula that gives 1 with probability 0.35, otherwise 0.

A simulation gave 128 successes in 500 trials. Estimate the probability. How could you improve the estimate?

Events as Sets and the Complement of an Event

S = {1, …, 8}, A = {numbers greater than 5}. Find A′.

A die is rolled 5 times. Find P(at least one 6).

Each of 8 light bulbs works with probability 0.98, independently. Find P(at least one fails).

Venn Diagrams: Intersection, Union and Counting

n(A) = 34, n(B) = 21, n(A ∩ B) = 9. Find n(A ∪ B).

In a group of 80, 50 like hockey, 35 like basketball, and 15 like both. How many like neither?

Three sets: n(A ∩ B ∩ C) = 2 and n(B ∩ C) = 7. How many are in B and C but not A?

Mutually Exclusive and Non-Mutually Exclusive Events

One card is drawn. Find P(a club or a face card).

Two dice are rolled. Find P(sum is 11 or doubles).

P(A) = 0.35, P(B) = 0.45, P(A ∪ B) = 0.8. Are A and B mutually exclusive?

Independent and Dependent Events

Two cards are drawn without replacement. Find P(both are red).

P(A) = 0.3, P(B) = 0.6, P(A ∩ B) = 0.18. Independent?

A bag has 2 red and 3 green marbles. Two are drawn without replacement. Find P(one of each colour).

A coin is tossed and a die is rolled. Find P(tails and a number greater than 4).

Conditional Probability

A die is rolled. Find P(roll is 1 | roll is less than 4).

P(A) = 0.4 and P(A ∩ B) = 0.1. Find P(B | A).

Of 150 students, 90 are in a club; 54 of the club members also play a sport. Find P(sport | club).

Solving Multi-Step Probability Problems

Plant X supplies 25% of a store's apples, and 8% of them are bruised; plant Y supplies 75%, and 4% are bruised. Find P(bruised).

Using the apple question, find P(from X | bruised).

A test for a condition that affects 2% of people gives a positive result 90% of the time when the condition is present and 5% of the time when it is absent. Out of 1000 people, how many positives are true positives?

Check your understanding

Quiz

Which value cannot be a probability?

How many outcomes does the sample space for tossing 3 coins and rolling 1 die have?

Two dice are rolled. What is P(sum is 10)?

The odds in favour of an event are 2 : 9. What is the probability?

A die is rolled 6000 times. Which statement is most reasonable?

A die is rolled 3 times. What is P(no 6 at all)?

P(A) = 0.5, P(B) = 0.3, P(A ∩ B) = 0.1. What is P(A ∪ B)?

Which pair of events is mutually exclusive for one roll of a die?

Two cards are drawn without replacement. What is P(both are kings)?

10.

In a survey, 120 people own a dog and 45 of them also own a cat. What is P(owns a cat | owns a dog)?

For parents and teachers

Use the review to find what to revisit:

Confusing kinds of probability, or probabilities outside 0 to 1 → Lesson 1 (Probability as a Measure of Likelihood).
Missing outcomes (HT vs TH), or unsure about discrete vs continuous → Lesson 2 (Sample Spaces).
Using n(A)/n(S) with outcomes that are not equally likely, or distributions not adding to 1 → Lesson 3 (Theoretical Probability and Probability Distributions).
Mixing up odds and probability, or odds in favour and against → Lesson 4 (Odds).
Believing results are "due", or trusting tiny experiments → Lesson 5 (Experimental Probability and the Law of Large Numbers).
Simulations with the wrong probabilities or too few trials → Lesson 6 (Simulations).
Adding instead of using 1 − P(none) for "at least one" → Lesson 7 (Complements).
Putting a set's total in its "only" region, or forgetting "neither" → Lesson 8 (Venn Diagrams).
Forgetting to subtract the overlap → Lesson 9 (Mutually Exclusive Events).
Treating draws without replacement as independent → Lesson 10 (Independent and Dependent Events).
Wrong denominator in a conditional probability → Lesson 11 (Conditional Probability).
Mixing up P(A | B) and P(B | A) in multi-step problems → Lesson 12 (Multi-Step Problems).

Looking ahead: the next unit, Permutations and Combinations, develops counting tools (factorials, permutations, combinations, Pascal's triangle) that make large sample spaces easy to count. Then Discrete Probability Distributions builds on the distributions and expected values introduced here.