165. Two-Step Probability Experiments

Learning Intentions

• Understand that a table can be used to list the sample space of a two-step experiment
• Calculate the probability of events in two-step experiments
• Calculate the probability of events in two-step experiments

Pre-requisite Summary

• A probability is a number between and
• An experiment is a chance process, and a trial is one performance of that process
• An outcome is one possible result, and an event is a set of outcomes
• The sample space is the set of all possible outcomes
• Fractions can be used to represent probabilities
• A table can organise outcomes clearly when an experiment has two steps
• To Solve a probability, compare the number of favourable outcomes with the total number of outcomes

Worked Examples

Worked Example 1

A coin is tossed and then a fair six-sided die is rolled.

Use a table to list the sample space.

Worked Example 2

A coin is tossed and then a fair six-sided die is rolled.

Find the probability of getting heads and a .

Worked Example 3

A coin is tossed and then a fair six-sided die is rolled.

Find the probability of getting a tail and an even number.

Worked Example 4

A spinner with equal sections labelled , and is spun, and then a coin is tossed.

Use a table to list the sample space and find the probability of getting and heads.

Worked Example 5

A card numbered , or is chosen, and then a fair coin is tossed.

Find the probability of getting an odd number and tails.

Worked Example 6

A fair die is rolled twice.

Use a table to list the sample space and find the probability of getting a total of .

Problems

Problem 1

A coin is tossed and then a fair six-sided die is rolled.

Use a table to list the sample space.

Problem 2

A coin is tossed and then a fair six-sided die is rolled.

Find the probability of getting tails and a .

Problem 3

A coin is tossed and then a fair six-sided die is rolled.

Find the probability of getting heads and an odd number.

Problem 4

A spinner with equal sections labelled , and is spun, and then a coin is tossed.

Use a table to list the sample space and find the probability of getting and tails.

Problem 5

A card numbered , or is chosen, and then a fair coin is tossed.

Find the probability of getting an even number and heads.

Problem 6

A fair die is rolled twice.

Use a table to list the sample space and find the probability of getting a total of .

Exercises

Understanding and Fluency

Exercise 1.

List the sample space for each two-step experiment.

a) toss a coin, then roll a fair six-sided die

b) spin a spinner labelled , , , then toss a coin

c) Choose a card numbered , , , then toss a coin

Exercise 2.

Use a table to list the sample space for each experiment.

a) roll a fair die, then toss a coin

b) toss a coin twice

c) choose a card numbered to , then toss a coin

Exercise 3.

A coin is tossed and then a fair six-sided die is rolled. Find each probability.

a)

b)

c)

Exercise 4.

A coin is tossed and then a fair six-sided die is rolled. Find each probability.

a)

b)

c)

Exercise 5.

A spinner with equal sections labelled , and is spun, and then a fair coin is tossed. Find each probability.

a)

b)

c)

Exercise 6.

A card numbered , , or is chosen, and then a fair coin is tossed. Find each probability.

a)

b)

c)

Exercise 7.

A fair die is rolled twice. Find each probability.

a) a total of

b) a total of

c) doubles

Exercise 8.

A fair die is rolled twice. Find each probability.

a) a total greater than

b) the first roll is and the second roll is

c) the two numbers have an odd total

Reasoning

Exercise 9.

Explain why a table is useful for listing the sample space of a two-step experiment.

Exercise 10.

A student says that tossing a coin and rolling a die gives outcomes because . Explain the error.

Exercise 11.

Explain why the probability of getting heads and a when tossing a coin and rolling a die is .

Exercise 12.

A student says that when a die is rolled twice, the outcomes and are the same. Explain why this is incorrect.

Exercise 13.

Explain why the total number of outcomes in a two-step experiment is often found by multiplying.

Problem-solving

Exercise 14.

A game uses a spinner with equal sections labelled , , and a fair coin.

Use a table to list the sample space, then find the probability of getting an even number and tails.

Exercise 15.

A bag contains cards labelled , , , . One card is chosen at random and then a fair coin is tossed.

Find the probability of getting a vowel and heads.

Exercise 16.

A fair die is rolled twice.

Find the probability of getting a total of , and list all favourable outcomes.

Exercise 17

A fair coin is tossed twice.

Find the probability of getting exactly one head.

Exercise 18.

A spinner has equal sections labelled red, blue and yellow. It is spun twice.

Find the probability of getting the same colour both times.

Potential Misunderstandings

• Thinking the total number of outcomes in a two-step experiment is found by adding instead of multiplying
• Forgetting that order matters in many two-step experiments, such as rolling a die twice
• Missing outcomes when listing a sample space without Use a table
• Counting some outcomes twice and others not at all
• Confusing a single outcome with an event made of several outcomes
• Using the number of favourable outcomes as the denominator instead of the total number of outcomes
• Thinking and are always the same outcome
• Forgetting that a table should include every possible result from the first step paired with every possible result from the second step
• Assuming all events in a two-step experiment have the same number of favourable outcomes

Next: 166. Tree Diagrams for Multi-Step Probability