074. Probability Language and Simple Events

Learning Intentions

• To know the meaning of the terms experiment, trial, outcome, event and sample space
• To understand that the probability of an event is a number between $0$ and $1$ inclusive representing the chance it will occur
• calculate probabilities of simple events

Pre-requisite Summary

• Understand that chance can be described in everyday language such as impossible, unlikely, even chance, likely and certain
• Be able to list possible results in simple chance situations such as tossing a coin or rolling a die
• Know that fractions can represent part of a whole
• Be able to count favourable outcomes and total outcomes
• Understand that a result can be recorded from one attempt or from many repeated attempts
• Know that a number line can be used to place values between $0$ and $1$
• Understand that $0$ means no chance and $1$ means certainty

Worked Examples

Worked Example 1

For a fair coin toss:
a) state the experiment
b) state one trial
c) list the sample space
d) identify one outcome and one event

Worked Example 2

For rolling a fair six-sided die:
a) list the sample space
b) identify the event “rolling an even number”
c) find the probability of rolling an even number

Worked Example 3

A bag contains $3$ red counters and $2$ blue counters. One counter is chosen at random.
a) list the possible outcomes by colour
b) find the probability of choosing a red counter
c) find the probability of choosing a blue counter

Worked Example 4

For spinning a fair spinner with equal sections labelled $A,B,C,D$:
a) identify the event “landing on a vowel”
b) find the probability of that event
c) place the probability on a scale from $0$ to $1$

Worked Example 5

A bag contains $5$ green marbles, $1$ yellow marble and $4$ purple marbles. One marble is chosen at random.
a) find the probability of choosing yellow
b) find the probability of choosing green
c) state which event is more likely

Worked Example 6

For a fair die:
a) find the probability of rolling a number greater than $4$
b) find the probability of rolling a number less than $7$
c) explain why one probability is close to $0$ and the other is equal to $1$

Problems

Problem 1

For a fair coin toss:
a) state the experiment
b) state one trial
c) list the sample space
d) identify one outcome and one event

Problem 2

For rolling a fair six-sided die:
a) list the sample space
b) identify the event “rolling an odd number”
c) find the probability of rolling an odd number

Problem 3

A bag contains $4$ red counters and $3$ blue counters. One counter is chosen at random.
a) list the possible outcomes by colour
b) find the probability of choosing a red counter
c) find the probability of choosing a blue counter

Problem 4

For spinning a fair spinner with equal sections labelled $A,E,I,O$:
a) identify the event “landing on a vowel”
b) find the probability of that event
c) place the probability on a scale from $0$ to $1$

Problem 5

A bag contains $6$ green marbles, $2$ yellow marbles and $2$ purple marbles. One marble is chosen at random.
a) find the probability of choosing yellow
b) find the probability of choosing green
c) state which event is more likely

Problem 6

For a fair die:
a) find the probability of rolling a number greater than $5$
b) find the probability of rolling a number less than $7$
c) explain why one probability is small and the other is equal to $1$

Exercises

Understanding and Fluency

1. Match each term with its meaning:
   a) experiment
   b) trial
   c) outcome

2. Match each term with its meaning:
   a) event
   b) sample space
   c) probability

3. For each chance situation, list the sample space:
   a) tossing a fair coin
   b) rolling a fair six-sided die
   c) spinning a fair spinner with sections $R,G,B$

4. Identify the event and favourable outcomes:
   a) rolling an even number on a die
   b) tossing a head on a coin
   c) choosing a vowel from cards labelled $A,B,C,E$

5. Calculate each probability:
   a) head on a fair coin
   b) rolling a $3$ on a fair die
   c) rolling an odd number on a fair die

6. Calculate each probability:
   a) choosing a red counter from a bag with $2$ red and $3$ blue
   b) choosing a blue counter from the same bag
   c) choosing a colour that is not in the bag

7. A spinner has equal sections labelled $1,2,3,4,5$. Find the probability of:
   a) landing on $2$
   b) landing on an odd number
   c) landing on a number greater than $3$

8. A bag contains $3$ green, $4$ red and $1$ yellow marble. Find the probability of:
   a) green
   b) yellow
   c) not yellow

9. State whether each probability is impossible, unlikely, even chance, likely or certain:
   a) $0$
   b) ${\displaystyle \frac{1}{2}}$
   c) $1$

10. State whether each probability is impossible, unlikely, even chance, likely or certain:
    a) ${\displaystyle \frac{1}{6}}$
    b) ${\displaystyle \frac{5}{6}}$
    c) ${\displaystyle \frac{3}{6}}$

Reasoning

11. Explain why the probability of an event must be between $0$ and $1$ inclusive.

12. A student says the probability of rolling a $7$ on a six-sided die is ${\displaystyle \frac{1}{7}}$. Explain the mistake.

13. Explain why the sample space for a fair coin toss has two outcomes.

14. A student says the probability of rolling a number less than $7$ on a six-sided die is ${\displaystyle \frac{6}{7}}$. Explain why this is incorrect.

15. Explain why an event with probability $1$ is certain.

16. A student says an outcome and an event are always the same thing. Explain why this is not always true.

Problem-solving

17. A bag contains $5$ black pens and $3$ blue pens. One pen is chosen at random.
    a) state the sample space by colour
    b) find the probability of choosing a blue pen
    c) state whether choosing a blue pen is unlikely, even chance, or likely

18. A fair spinner has equal sections labelled $A,B,C,D,E$.
    a) find the probability of landing on a vowel
    b) find the probability of landing on a consonant
    c) state which event is more likely

19. A fair die is rolled once.
    a) find the probability of rolling a factor of $6$
    b) find the probability of rolling a number greater than $2$
    c) compare the two probabilities

20. A box contains $2$ red marbles, $2$ green marbles and $6$ white marbles. One marble is chosen at random.
    a) find the probability of choosing green
    b) find the probability of choosing red or green
    c) state whether choosing white is likely or unlikely

Potential Misunderstandings

• Students may confuse an experiment with a trial
• Students may think an outcome and an event always mean exactly the same thing
• Students may list outcomes incorrectly when forming a sample space
• Students may count all outcomes incorrectly when calculating probability
• Students may forget that probability is a number from $0$ to $1$ inclusive
• Students may think a larger denominator always means a larger probability
• Students may use the number of favourable outcomes as the probability without dividing by the total number of outcomes
• Students may think impossible events can have a small positive probability instead of $0$

Next: 075. Experimental Probability and Expected Frequency