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 $0$ and $1$
• 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 find 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 $4$.

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 $A$, $B$ and $C$ is spun, and then a coin is tossed.
Use a table to list the sample space and find the probability of getting $B$ and heads.

Worked Example 5
A card numbered $1$, $2$ or $3$ 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 $7$.

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 $5$.

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 $R$, $G$ and $B$ is spun, and then a coin is tossed.
Use a table to list the sample space and find the probability of getting $G$ and tails.

Problem 5
A card numbered $1$, $2$ or $3$ 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 $8$.

Exercises

Understanding and Fluency

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 $A$, $B$, $C$, then toss a coin
   c) choose a card numbered $1$, $2$, $3$, then toss a coin

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 $1$ to $4$, then toss a coin

3. A coin is tossed and then a fair six-sided die is rolled. Find each probability.
   a) $P(\text{H and }3)$
   b) $P(\text{T and }6)$
   c) $P(\text{H and even})$

4. A coin is tossed and then a fair six-sided die is rolled. Find each probability.
   a) $P(\text{T and odd})$
   b) $P(\text{H and number greater than }4)$
   c) $P(\text{T and number less than }3)$

5. A spinner with equal sections labelled $R$, $G$ and $B$ is spun, and then a fair coin is tossed. Find each probability.
   a) $P(R\text{ and H})$
   b) $P(B\text{ and T})$
   c) $P(G\text{ and H})$

6. A card numbered $1$, $2$, $3$ or $4$ is chosen, and then a fair coin is tossed. Find each probability.
   a) $P(2\text{ and H})$
   b) $P(\text{odd and T})$
   c) $P(\text{number greater than }2\text{ and H})$

7. A fair die is rolled twice. Find each probability.
   a) a total of $5$
   b) a total of $12$
   c) doubles

8. A fair die is rolled twice. Find each probability.
   a) a total greater than $9$
   b) the first roll is $2$ and the second roll is $4$
   c) the two numbers have an odd total

Reasoning

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

10. A student says that tossing a coin and rolling a die gives $8$ outcomes because $2+6=8$. Explain the error.

11. Explain why the probability of getting heads and a $2$ when tossing a coin and rolling a die is ${\displaystyle \frac{1}{12}}$.

12. A student says that when a die is rolled twice, the outcomes $(2,5)$ and $(5,2)$ are the same. Explain why this is incorrect.

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

Problem-solving

14. A game uses a spinner with equal sections labelled $1$, $2$, $3$ and a fair coin.

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

15. A bag contains cards labelled $A$, $B$, $C$, $D$. One card is chosen at random and then a fair coin is tossed.

    Find the probability of getting a vowel and heads.

16. A fair die is rolled twice.

    Find the probability of getting a total of $10$, and list all favourable outcomes.

17. A fair coin is tossed twice.

    Find the probability of getting exactly one head.

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 using 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 $(3,5)$ and $(5,3)$ 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