The probability scale
🎯 In this topic you will
- Use the language associated with probability
- Understand how probabilities range from 0 to 1
🧠 Key Words
- event
- even chance
- likely
- likelihood
- outcome
- probability
- unlikely
Show Definitions
- event: A specific outcome or set of outcomes in a probability experiment.
- even chance: When an event has the same probability of happening as not happening (50%).
- likely: Describes an event with a high probability of occurring.
- likelihood: Another word for the probability of an event happening.
- outcome: The result of a single trial in a probability experiment.
- probability: A measure of how likely an event is to occur, expressed between 0 and 1.
- unlikely: Describes an event with a low probability of occurring.
🎲 Events and Outcomes
Rolling an odd number on a dice or winning a football match are examples of events.
Events can have a number of different outcomes.
The outcomes that give an odd number when you roll a dice are $1, 3$ and $5$.
The outcomes when you win a football match are the possible scores.
You can use words such as certain, likely, unlikely and impossible to describe the likelihood that an event will happen.
You can also use a number between $0$ and $1$ to represent the likelihood that an event will happen. This number is called a probability.
An event that is certain to happen has a probability of $1$.
An outcome that is impossible has a probability of $0$.
You can show a probability on a probability scale.
You can write a probability as a fraction, a decimal or a percentage.
For example, an even chance means a probability of $0.5$ or $\tfrac{1}{2}$ or $50\%$.
❓ EXERCISES
1. Here are some words that describe likelihood.
a. Find some outcomes that can be described by each of these words or phrases.
b. Can you think of any other words or phrases to describe likelihood? If you can, give some examples.
👀 Show answer
a. Examples:
- $\text{likely}$: It will rain sometime this month.
- $\text{unlikely}$: Winning a lottery.
- $\text{very likely}$: The sun will rise tomorrow.
- $\text{very unlikely}$: Finding a diamond on the street.
- $\text{certain}$: The day after Monday is Tuesday.
- $\text{impossible}$: Rolling a $7$ on a fair die.
- $\text{even chance}$: Flipping a coin gives heads.
b. Other possible words: “almost certain,” “doubtful,” “possible.”
2. Choose the best word or phrase to describe these events.
a. When you flip a coin, it will land showing heads.
b. The day after Monday will be Tuesday.
c. You have the same birthday as your teacher.
d. It will rain one day next week.
e. You will do well in your next maths test.
👀 Show answer
- a. Even chance
- b. Certain
- c. Very unlikely
- d. Likely
- e. Unlikely
❓ EXERCISES
3. You throw a dice. Put these events (A to E) in order of likelihood. Put the least likely first.
A You throw the number $3$.
B You throw the number $3$ or more.
C You throw a number less than $3$.
D You throw an odd number.
E You throw a number less than $1$.
👀 Show answer
Order: E (impossible), A (1 out of 6), C (2 out of 6), D (3 out of 6), B (4 out of 6).
4. The probability of rain tomorrow is $25\%$.
The probability of sunny weather tomorrow is $60\%$.
a. Write both probabilities as fractions.
b. Show each weather event’s probability on a probability scale.
👀 Show answer
a. Rain: $\tfrac{25}{100} = \tfrac{1}{4}$, Sunny: $\tfrac{60}{100} = \tfrac{3}{5}$.
b. Mark $\tfrac{1}{4}$ and $\tfrac{3}{5}$ on the probability scale between 0 and 1.
5. Here are the probabilities that three teams will win their next match:
City $\tfrac{2}{3}$, Rovers $60\%$, United $0.7$
a. Which team is most likely to win? Give a reason for your answer.
b. Which team is least likely to win?
👀 Show answer
a. City has $\tfrac{2}{3} \approx 66.7\%$, Rovers $60\%$, United $70\%$. United has the highest probability, so they are most likely to win.
b. Rovers $60\%$ is the lowest, so they are least likely to win.
6. Here is a probability scale.
Estimate the probability of each event.
👀 Show answer
Answers will vary depending on the estimated positions of P, Q, R, S. Example: P ≈ 0.1, Q ≈ 0.3, R ≈ 0.6, S ≈ 0.9.
7.
a. Draw a probability scale. Mark these events on the diagram.
A Zhing will be late for school: $25\%$
B Rain will fall in the town: $0.6$
C The football match will be a draw: $\tfrac{1}{5}$
D A plant will flower: $80\%$
E Roshni will study maths at university: $\tfrac{9}{10}$
F The train will be late: $0.05$
b. Compare your probability scale with a partner’s.
👀 Show answer
a. Plot each probability at the correct position: A at 0.25, B at 0.6, C at 0.2, D at 0.8, E at 0.9, F at 0.05.
b. Answers will vary depending on scale accuracy but should match the given values.
8.
a. Suggest some events that could have the following probabilities.
i $ \tfrac{1}{10} $ ii $50\%$ iii $0.85$ iv $100\%$
👀 Show answer
- i. Rolling a 6 on a biased 10-sided spinner.
- ii. Tossing a fair coin.
- iii. The sun rising tomorrow.
- iv. Tuesday will follow Monday.
🧠 Think like a Mathematician
9a. A weather forecast says:
How do you think this probability was worked out?
9b. Try to find some examples of probabilities being used in the news. How do you think the probabilities were worked out?
👀 Show Answer
- 9a: Weather forecasts use computer models that combine historical data, current weather patterns, and probability calculations. An $80\%$ chance of rain means that in similar conditions, rain occurred about 80 times out of 100.
- 9b: Examples include election polling percentages, chances of a team winning a sports match, or risk levels in health studies. These are usually worked out using statistical models, large data samples, or simulations.
❓ EXERCISES
10. Sogand flips a coin. The probability that it lands with heads facing up is $50\%$.
Sogand flips the coin $5$ times. She gets heads every time.
Look at these three statements.
A The probability that the next flip lands with heads facing up is $50\%$.
B The probability that the next flip shows heads is more than $50\%$.
C The probability that the next flip shows heads is less than $50\%$.
Which statement do you think is correct? Give a reason for your answer.
👀 Show answer
Correct statement: A. For a fair coin, each flip is independent, so the probability of heads on the next flip remains $50\%$ regardless of previous results. The streak of $5$ heads does not change the probability (avoids the “gambler’s fallacy”).