Probability

Probability of Compound Events

Probability Terms

Draw a line to match each term to its meaning.

Compound event

Independent events

Complementary events

Sample space

The outcome of one does not affect the other

P(A) + P(B) = 1

The set of all possible outcomes

An event made up of two or more simple events

Two-Way Tables

Class of 30: 12 boys, 18 girls. 8 boys and 10 girls like sport.

P(girl who likes sport):

10/30 = 1/3

10/18

18/30

P(someone who likes sport):

18/30 = 3/5

10/30

8/30

P(boy who does NOT like sport):

4/30 = 2/15

8/30

12/30

Probability from Sample Spaces

A coin is flipped and a die is rolled.

Total outcomes in sample space:

P(heads AND a 6):

1/12

1/6

1/2

P(tails AND an even number):

3/12 = 1/4

1/2

1/6

Independent or Dependent?

Sort each pair of events.

Flip a coin; roll a die

Draw a card; draw another WITHOUT replacing

Pick a marble; replace it; pick again

Pick a marble without replacing; pick again

Roll two dice

Select a student; then select another (no replacement)

Independent

Dependent

Multiply for Independent Events

P(A and B) = P(A) × P(B) for independent events.

P(heads) × P(rolling a 3)

1/12

1/6

1/3

P(two sixes in a row):

1/36

1/12

1/6

P(girl born twice if P(girl) = 0.5):

0.25

0.5

0.1

Experimental vs Theoretical

Sort each statement.

Calculated from equally likely outcomes

Found by running many trials

P(head) = 1/2 because there are 2 sides

After 200 flips, heads came up 104 times

Approaches theoretical as trial count increases

Theoretical

Experimental

Compound Event Problems

Show all working. Draw a table or tree diagram.

A spinner has 3 equal sections (Red, Blue, Green). Spun twice. List all 9 outcomes. What is P(Red both times)?

Draw here

A bag has 4 red and 6 blue balls. Draw one, replace it, draw again. What is P(blue then red)?

P(A or B) for Mutually Exclusive Events

Mutually exclusive events cannot happen at the same time. P(A or B) = P(A) + P(B).

Roll a die. P(rolling a 2 or a 5):

2/6 = 1/3

1/36

5/6

P(red card or black card) in a standard deck:

1/2

1/4

Bag: 3 red, 4 blue, 5 green. P(red or blue):

7/12

3/12

12/12

Venn Diagram Probabilities

Use the Venn diagram information to answer each question.

30 students: 18 play sport, 12 play music, 6 play both. Draw a Venn diagram. How many play sport only? Music only? Neither?

Draw here

From the diagram, find P(a random student plays sport or music). Show your working using P(A or B) = P(A) + P(B) − P(A and B).

Conditional Probability Introduction

P(B given A) means the probability of B, knowing A has already happened.

A bag has 3 red and 2 blue balls. You draw red first (no replacement). P(blue second) =

2/4 = 1/2

2/5

3/4

A class has 10 boys and 10 girls. One person is chosen (a girl). P(next person is also a girl) =

9/19

10/19

10/20

P(A) = 0.4, P(B|A) = 0.5. P(A and B) =

0.2

0.4

0.9

Which Probability Rule Applies?

Sort each problem by the rule you would use.

P(flip heads OR roll a 3)

P(roll a 2 OR roll a 5) on one die

P(likes sport OR likes art) when some like both

P(flip heads AND flip heads again)

P(pick red OR pick blue) from one bag

P(roll even OR roll a 3) — 3 is not even

P(A) + P(B): mutually exclusive

P(A) × P(B): independent

P(A) + P(B) − P(A and B): overlapping

Two-Way Tables — Construction

Complete the two-way frequency table and answer the questions.

Survey of 50 students: 30 like sport, 20 don't. Of those who like sport, 18 are boys. Of those who don't, 6 are boys. Complete a two-way table (rows: Like sport / Don't like sport; columns: Boy / Girl).

Draw here

What is P(a randomly chosen student is a girl who likes sport)?

What is P(a randomly chosen student likes sport)?

TipTwo-way tables organise data by two categorical variables simultaneously.

Tree Diagram — Two Events

Draw a tree diagram and list the sample space.

A bag contains 2 red and 3 blue balls. Draw one ball, replace it, then draw another. Draw the tree diagram and list all outcomes.

Draw here

P(red then blue) = ?

P(same colour both times) = ?

TipEncourage neat tree diagrams with probability labels on each branch.

Tree Diagram — Three Events

Draw a tree diagram for three coin flips.

List all 8 outcomes of flipping a coin three times.

Draw here

P(exactly two heads) = ?

P(at least one head) = ?

TipA three-stage tree will have 2 × 2 × 2 = 8 branches at the end.

Sample Space Tables

Complete each two-way table to show all outcomes, then answer the probability questions.

Draw a table for rolling two dice. List all 36 outcomes. How many ways can you get a sum of 7?

Draw here

P(sum of 7) = ? P(sum of 12) = ?

P(sum is even) = ? (Count even sums in your table.)

TipTables work well when both events have a small number of outcomes.

P(A or B) for Non-Mutually Exclusive Events

Use P(A or B) = P(A) + P(B) − P(A and B) when events can overlap.

From a standard deck of 52 cards, what is P(drawing a king OR a heart)? Show working. (Note: king of hearts is in both.)

In a class of 30: 18 play sport, 12 do art, 6 do both. P(sport or art) = ?

TipThis formula prevents double-counting outcomes in both A and B.

Without Replacement — Dependent Events

When items are NOT replaced, each draw affects the next. Show all working.

A bag has 4 red and 6 blue balls. Draw two without replacement. P(both red) = ?

From 52 cards, draw two without replacement. P(both aces) = ?

From 52 cards, draw two without replacement. P(first ace, second king) = ?

TipAfter drawing one item, there is one fewer item in the bag — so the denominator decreases.

With Replacement or Without?

Sort each scenario: with replacement (independent) or without replacement (dependent).

Roll a die twice

Select a student; they leave; select another

Pick a card, put it back, pick again

A quality control inspector checks items from a batch, setting aside each one

Flip a coin three times

Select names from a hat without putting them back

With replacement (independent)

Without replacement (dependent)

Venn Diagram Probabilities

Use a Venn diagram to organise and calculate probabilities.

100 people: 60 like cats, 45 like dogs, 20 like both. Draw a Venn diagram. How many like neither?

Draw here

P(likes cats or dogs) = ? P(likes cats but not dogs) = ?

TipVenn diagrams are ideal for 'or' probability problems with overlapping events.

Spinner Experiments

Draw tree diagrams and calculate probabilities.

Spinner A has sections Red (1/2), Blue (1/4), Green (1/4). Spinner B has sections Yellow (1/3), Purple (2/3). Spin both. P(Red and Yellow) = ?

P(not Red and not Purple) = ? (Use complements.)

TipEncourage labelling branches with fractions, not percentages.

Experimental Probability

Compare experimental and theoretical probabilities.

A die is rolled 60 times. Results: 1→8, 2→11, 3→9, 4→10, 5→12, 6→10. Calculate the experimental probability of rolling each number.

Compare to theoretical probability (1/6 each). Are the results consistent with a fair die?

How many trials would you need for experimental probability to be reliably close to 1/6?

TipExperimental probability approaches theoretical probability as the number of trials increases (Law of Large Numbers).

Simulation Design

Design a simulation to estimate each probability.

Design a coin-flipping simulation to estimate P(exactly two heads in three flips). Describe the simulation and what you would count.

How many trials would give a reliable estimate? Why?

TipSimulations allow experimental verification of theoretical probability without performing the actual event.

Pascal's Triangle and Probability

Pascal's triangle shows the number of ways to get each outcome in coin-flipping experiments.

Write out rows 0 through 4 of Pascal's triangle.

Draw here

Use row 4 to find P(exactly 2 heads in 4 coin flips). Show working.

What is P(all 4 heads)? P(at least 1 head)?

TipRow n of Pascal's triangle gives the number of ways to get 0, 1, 2, ... n heads in n flips.

Probability in Games

Apply probability to a board game scenario.

In a game, you roll two dice. You win if the sum is 7 or 11. Find P(winning on one roll). Use a 6×6 table.

Draw here

P(losing on one roll) = ? P(winning exactly once in two rolls) = ?

TipGame probability problems are engaging and apply equally to board games and card games.

Conditional Probability Problems

P(B|A) means the probability of B given A has occurred.

A card is drawn from 52. Given it is a red card, what is P(it is a heart)? Explain why the sample space changes.

10 red and 10 blue balls in a bag. One is drawn and found to be red (not replaced). What is P(second draw is also red)?

TipConditional probability reduces the sample space to only the outcomes where A occurred.

Cross-Strand: Probability and Statistics

Connect probability to statistical investigation.

Theory predicts 1/6 ≈ 16.7% of rolls should be each number on a fair die. You roll 120 times and get: 1→18, 2→22, 3→17, 4→21, 5→20, 6→22. Is the die fair? Use percentages to compare experimental and theoretical frequencies.

TipProbability predicts what SHOULD happen. Statistics measures what DID happen. The two fields are deeply connected.

Error Analysis

Find and correct the errors in this student's working.

Student: 'P(Head) = 0.5 and P(6) = 1/6. P(Head and 6) = 0.5 + 1/6 = 0.667.' What is wrong? What is the correct answer?

Student: 'P(drawing 2 aces from 52 cards) = 4/52 × 4/52 = 16/2704.' What is wrong? What is the correct answer?

TipLearning to identify errors is as important as solving problems correctly.

Designing a Fair Game

Use probability to design a fair game.

Two players roll a die. Player A wins if the result is 1, 2, or 3. Player B wins if the result is 4, 5, or 6. Is this fair? Calculate each player's probability.

Design a different game using two dice where each player has P(win) = 1/2. Explain how you designed it.

Draw here

TipA fair game is one where each player has an equal probability of winning.

Expected Frequency

Use probability to predict frequencies.

A die is rolled 600 times. How many times would you expect to roll each number?

Two coins are flipped 200 times. How many times would you expect HH, HT, TH, TT each?

In a class of 25 students, each has a 70% chance of passing a test. How many students would you expect to pass?

TipExpected frequency = probability × number of trials. This connects theoretical probability to real-world data.

Probability Problem Solving

Show all working for each problem.

A teacher randomly selects 2 students from a group of 5 (3 boys: Ali, Ben, Carl; 2 girls: Dani, Eve). List all possible pairs. What is P(both girls)?

Draw here

What is P(one boy and one girl in the selected pair)?

TipEncourage drawing a table or tree diagram for each problem before calculating.

Reflection and Summary

Answer each question to consolidate your learning.

List three tools you can use to find sample spaces for compound events. When would you use each one?

Explain the difference between independent and dependent events. Give one example of each.

What is the difference between P(A and B) and P(A or B)? Give formulas and an example of each.

Draw here

TipMathematical reflection develops metacognition and identifies gaps for revision.

Applied Probability: Genetics

Each parent passes one gene randomly. Each gene is either dominant (B) or recessive (b).

Parent 1 is Bb and Parent 2 is Bb. Draw a 2×2 table (Punnett square) showing all possible offspring gene combinations. List all outcomes.

Draw here

If B is dominant, what is P(offspring shows dominant trait)? What is P(offspring is Bb)?

TipThis real-world application of compound probability appears in biology. It is a natural cross-strand connection.

Extended Investigation: Dice Sums

Investigate the distribution of sums when two dice are rolled.

Complete the 6×6 sum table for two dice. Count the number of ways to get each sum from 2 to 12.

Draw here

Which sum has the highest probability? What is P(sum = 7)? P(sum = 2 or 12)?

Calculate the expected sum (multiply each sum by its probability and add). What is the expected value?

TipThis investigation reveals why 7 is the most common sum — a fundamental result in probability.

Compound Probability Experiment

Try these experiments at home.

1Flip 2 coins simultaneously. Record outcomes (HH, HT, TH, TT) for 40 trials. Compare to theoretical probabilities.
2Roll 2 dice and add totals. Record for 36 rolls. Which total appeared most? Compare to theoretical distribution.
3Make a spinner with 3 coloured sections. Spin twice, record results. Calculate P(matching colours) experimentally and theoretically.