Describing & predicting likelihood
🎯 In this topic you will
- Describe the chance of outcomes using fractions and percentages.
- Learn about events that are mutually exclusive.
- Use likelihood to predict outcomes.
- Conduct chance experiments and describe the results.
🧠 Key Words
- event
- equally likely outcomes
- mutually exclusive events
- outcome
- probability
- probability experiment
Show Definitions
- event: A set of one or more outcomes from a probability experiment.
- equally likely outcomes: Outcomes that all have the same chance of occurring.
- mutually exclusive events: Events that cannot happen at the same time.
- outcome: A single possible result of a probability experiment.
- probability: A measure of how likely an event is to occur, often written as a fraction or percentage.
- probability experiment: An action or process that leads to one of several possible outcomes.
🎯 Predicting Chances
If we can understand and describe the likelihood of different events occurring, then we can predict how likely they are to occur in the future.
📊 Describing Likelihood Precisely
We can use fractions and percentages to describe likelihood more precisely than words.
❓ EXERCISES
1. Copy and complete the sentences about the balls in the bag.
a. The probability of a red ball being pulled from the bag is $\frac{1}{4}$.
b. The probability of a yellow ball being pulled from the bag is $\frac{1}{4}$.
c. The probability of a green ball being pulled from the bag is $\frac{2}{4}$ or $\frac{1}{2}$.
👀 Show answer
b. $\frac{1}{4}$
c. $\frac{2}{4}$ or $\frac{1}{2}$
2. Write the probability of each of these events occurring as a fraction and as a percentage.
a. taking a red card
b. taking a $2$
c. taking a card with a value higher than $4$
d. taking a card that is not a $3$
👀 Show answer
b. $\frac{1}{10} = 10\%$
c. $\frac{4}{10} = 40\%$
d. $\frac{9}{10} = 90\%$
3. Draw sets of cards that match the descriptions.
a. There is a $25\%$ chance of taking an $8$.
b. The probability of taking a card with a value less than $5$ is $\frac{5}{6}$.
c. The chance of taking a $3$ is greater than the chance of taking a $1$.
d. There is a $70\%$ chance of taking a card with a value greater than $4$.
e. There is a $2$ out of $5$ chance of taking a $3$.
👀 Show answer
4. Keran flips a coin and records whether the coin lands heads up, or tails up.
These are her results:
a. How many trials did Keran carry out?
b. Does Keran’s experiment show that the coin is more likely to land heads up or tails up?
c. What does Keran’s experiments show is the experimental probability of the coin landing tails up?
👀 Show answer
b. Heads up is more likely
c. $\frac{16}{40} = \frac{2}{5} = 40\%$
❓ EXERCISES
5. Some children play a game with numbered tickets from $1$ to $30$. They take a ticket without looking. If their number is odd they win a small prize. If their number is a multiple of $10$ they win a medium prize. If their number is both odd and a multiple of $10$ then they win a big prize.
Copy and complete this Venn diagram with the numbers $1$ to $30$.
a. Shade red the section of the diagram with the numbers that would not win a prize.
b. Shade blue the section of the diagram with the numbers that would win a small prize.
c. Shade yellow the section of the diagram with the numbers that would win a medium prize.
d. What is the chance of winning a big prize? Why?
e. Are the events ‘taking an odd number’ and ‘taking a multiple of $10$’ mutually exclusive?
👀 Show answer
Multiples of $10$: $10, 20, 30$
Intersection: none
Outside both sets: $2, 4, 6, 8, 12, 14, 16, 18, 22, 24, 26, 28$
a. Shade the outside region
b. Shade the odd-number region only
c. Shade the multiples-of-$10$ region only
d. The chance of winning a big prize is $0$ because no number from $1$ to $30$ is both odd and a multiple of $10$
e. Yes, these events are mutually exclusive
6. Look at the shapes in this bag.
Write ‘mutually exclusive’ or ‘not mutually exclusive’ for these pairs of events.
a. Taking shape that is red and taking a shape that is blue.
b. Taking shape that is red and taking a shape that is a cone.
c. Taking a shape with more than $5$ faces and taking a shape that is blue.
👀 Show answer
b. Not mutually exclusive
c. Mutually exclusive
7. You have two six-sided dice. One is red and one is blue.
Imagine you are going to investigate these events:
Event A: You roll a double.
Event B: The sum of the two scores is even.
Event C: The score on the blue dice is greater than the score on the red dice.
Event D: You get a $6$ on the red dice.
a. Which events can happen at the same time?
i. A and B?
ii. A and C?
iii. A and D?
iv. B and C?
v. B and D?
vi. C and D?
b. Which pairs of events are mutually exclusive?
c. Imagine you have already rolled the red dice. It is a $6$.
What is the chance of each of the event A, B, C and D happening now?
d. Write two events of your own about the dice that not mutually exclusive.
e. Write two events of your own about the dice that are mutually exclusive.
👀 Show answer
i. Yes
ii. No
iii. Yes
iv. Yes
v. Yes
vi. No
b. The mutually exclusive pairs are A and C, and C and D
c. If the red dice is already $6$:
Event A: $\frac{1}{6}$
Event B: $\frac{1}{2}$
Event C: $0$
Event D: $1$
d. Answers will vary
e. Answers will vary
8. Look at this spinner.
Asubi uses his knowledge of likelihood and fractions to predict that after $8$ spins the spinner is most likely to land on:
red $4$ times
blue $3$ times
yellow $1$ time.
Predict how times each colour will be landed on for these numbers of spins.
a. $16$ spins
b. $40$ spins
c. $200$ spins
👀 Show answer
b. Red $20$, blue $15$, yellow $5$
c. Red $100$, blue $75$, yellow $25$
9. Take a total of $10$ red and blue objects. For example, counters, cubes or beads.
a. How many blue objects?
b. How many red objects?
Hide the objects, for example in a bag or under a cloth of piece of paper.
You are going to take, record and replace an object $20$ times.
c. How many red objects would you expect to take?
d. How many blue objects would you expect to take?
e. Conduct the experiment. Record the colour of the counters you get in a tally chart.
f. Describe your results. Do your results match your prediction?
👀 Show answer
If the number of red objects is $r$ and the number of blue objects is $b$, where $r + b = 10$, then:
c. Expected red objects taken in $20$ draws = $20 \times \frac{r}{10} = 2r$
d. Expected blue objects taken in $20$ draws = $20 \times \frac{b}{10} = 2b$
e. Results will depend on the experiment
f. Results may be close to the prediction, but may not match exactly
🧠 Think like a Mathematician
Question: Vanessa makes this prediction:
There are $12$ different outcomes when I roll two dice and add the numbers.
The chance of rolling a $12$ is one out of twelve. I predict that if I roll two dice $60$ times I will most likely roll $12$ five times.
Method:
- Roll two dice and add the numbers.
- Use the table to record the tally for each total from $1$ to $12$.
- Complete the total column for each number rolled.
- Carry out Vanessa’s investigation and compare the results with the prediction.
- Discuss what happened in the investigation and write about what you found out.
| Number rolled | Tally | Total |
|---|---|---|
| $1$ | ||
| $2$ | ||
| $3$ | ||
| $4$ | ||
| $5$ | ||
| $6$ | ||
| $7$ | ||
| $8$ | ||
| $9$ | ||
| $10$ | ||
| $11$ | ||
| $12$ |
Follow-up Questions:
Show Answers
- 1: Vanessa’s prediction is not a very good prediction because the total $12$ does not have a chance of $\frac{1}{12}$. When two dice are rolled, $12$ can only happen in one way: $(6,6)$, so the chance is $\frac{1}{36}$.
- 2: The result of the experiment may be different from the prediction. In $60$ trials, you would usually expect the number $12$ only about $60 \times \frac{1}{36} \approx 1.7$ times, so about $1$ or $2$ times.
- 3: If you carried out more trials, the results would usually get closer to the true probabilities. That means the relative frequency of each total would become more stable.
- 4: A good conclusion is that not all totals from two dice are equally likely. Some totals can be made in many ways, but $12$ can only be made in one way, so it is less likely than Vanessa predicted.