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Likelihood

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visibility 76update 4 months agobookmarkshare

🎯 In this topic you will

Position the likelihood of events on a likelihood scale
Learn about equally likely events

🧠 Key Words

certain
equally likely
even chance
impossible
likely
outcome
unlikely

Show Definitions

certain: An event that will definitely happen; its probability is 1.
equally likely: When two or more outcomes all have the same probability of occurring.
even chance: An event that is as likely to happen as not happen; its probability is 0.5.
impossible: An event that cannot happen under any circumstances; its probability is 0.
likely: An event that has a high chance of occurring but is not guaranteed.
outcome: A possible result of a probability event or experiment.
unlikely: An event that has a low chance of occurring.

Using Probability to Make Decisions

P robability can help you choose the best time to do something by showing how likely different events are to happen.

Needing the Right Conditions

T his astronomer needs a clear night sky in order to observe the stars, so she checks the forecast to find the best time.

📘 Worked example

A. Draw arrows on the likelihood scale to show the chance of each event happening.

Rolling a 5 on a 6-sided dice.

B. Flipping a coin and getting heads.

Answer:

A. A 6-sided dice has six possible outcomes, and only one outcome is a 5. Therefore the probability of rolling a 5 is $\dfrac{1}{6}$, which is **unlikely**.

B. A coin has two equally likely outcomes, heads and tails. Therefore the probability of landing on heads is $\dfrac{1}{2}$, which is an **even chance**.

For event A, because only one out of six faces shows a 5, the chance is low. On the likelihood scale, the arrow should be placed in the “unlikely” region.

For event B, heads and tails are equally likely, giving a probability of one half. On the likelihood scale, the arrow should point to “even chance”.

The likelihood scale visually represents how certain or uncertain an outcome is, from impossible to certain.

❓ EXERCISES

1. Copy this likelihood scale.

Draw arrows to show the likelihood of the following events happening.

a. When you roll a dice you will get a $3$.

b. When you flip a coin it will land on tails.

c. It will rain today.

d. Write three statements of your own and add them to the likelihood scale.

👀 Show answer

a. Getting a $3$ on a fair dice has probability $1/6$, so it is unlikely.

b. A fair coin has an equal chance of heads or tails, so landing on tails is an even chance.

c. This depends on the weather conditions where you live. Without data, the likelihood cannot be determined.

d. Answers will vary. Example statements: “I will see a bird today.” (likely) “I will see a dinosaur today.” (impossible) “I will eat something sweet today.” (likely)

2. Copy this likelihood scale.

Look at this spinner.

Mark the likelihood of each of these outcomes on your likelihood scale.

a. Scoring an odd number.

b. Scoring an even number.

c. Scoring less than $5$.

d. Scoring a number greater than $6$.

👀 Show answer

The spinner has four equal sections labelled $7$, $9$, $3$ (all odd) and $5$ (also odd). There are no even numbers.

a. Odd numbers: $4/4$. This is certain.

b. Even numbers: $0/4$. This is impossible.

c. Less than $5$: only $3$, so $1/4$. This is unlikely.

d. Numbers greater than $6$: $7$ and $9$, so $2/4$. This is an even chance.

3. Which two shapes are equally likely to be taken from this bag?

👀 Show answer

The picture shows two cubes and two triangular-based solids. These two shapes appear the same number of times, so they are equally likely to be taken.

4. Sofia has this bag of letter tiles. She takes one tile out at a time, writes the letter, then puts it back.

Copy and complete these sentences:

a. There are _____ letter tiles.

b. _____ of the tiles are the letter E.

c. The chance of taking a letter E is unlikely / even chance / certain.

d. It is equally likely that Sofia will take letter _____ or letter _____.

e. Letter _____ is the most likely to be taken.

👀 Show answer

The bag contains the letters: $E, S, T, E, E, T, M, E$.

a. There are $8$ letter tiles.

b. $4$ of the tiles are the letter E.

c. $4/8 = 1/2$ so taking an E is an even chance.

d. Sofia is equally likely to take S or M because each appears once.

e. The most likely letter is E because it appears most often.

5. Write two sentences of your own about the likelihood of Sofia taking different tiles from the bag in question $4$.

👀 Show answer

Example answers (many others possible):

“It is unlikely that Sofia will take the letter M because there is only one M tile.”

“It is likely that she will take the letter E because there are more E tiles than any other letter.”

🧠 Think like a Mathematician

Scenario: Sarah and Lou are playing a game with two coins. You will investigate whether each player is equally likely to win.

Rules of the Game:

Two coins are flipped at the same time.
Sarah scores 1 point if both coins land heads up.
Lou scores 1 point if one coin is heads and the other is tails.
No points are scored if both coins land tails up.

Main Question: Do you think Sarah and Lou are equally likely to win?

Activity:

Draw a table to record the outcomes of the coin flips:

Tally	Total
Sarah’s score
Lou’s score

Flip two coins 50 times and record your results in the table.
Decide whether your results suggest that Sarah and Lou are equally likely to win.
Write your thoughts using the language of chance.

Helpful Thinking:

You are conjecturing when you form an idea about whether Sarah and Lou are equally likely to win.
You are convincing when you use your recorded results to explain your conclusion.

Follow-up Questions:

1. List all possible outcomes when flipping two coins.

2. What is the probability that Sarah scores a point?

3. What is the probability that Lou scores a point?

4. Based on these probabilities, who is more likely to win the game?

Show Answers

1: The four possible outcomes are: HH, HT, TH, TT.
2: Sarah scores when both coins are heads. Probability = $\frac{1}{4}$.
3: Lou scores when one coin is heads and one is tails. Outcomes: HT, TH. Probability = $\frac{2}{4} = \frac{1}{2}$.
4: Lou is more likely to win because $\frac{1}{2} > \frac{1}{4}$. Lou scores twice as often as Sarah.

📘 What we've learned

We learned how to place events on a likelihood scale ranging from "impossible" to "certain".
We used everyday situations (coin flips, dice rolls, weather predictions, spinners, and bags of objects) to compare the chance of events happening.
We explored when two events are equally likely, such as two shapes appearing the same number of times in a bag.
We identified events with even chance, such as getting heads or tails when flipping a fair coin.
We worked out probabilities using simple ratios, such as the chance of rolling a specific number on a $6$-sided dice: $\frac{1}{6}$.
We analysed two-coin outcomes to see which player is more likely to score in a game: Sarah scores on HH → $\frac{1}{4}$, Lou scores on HT or TH → $\frac{2}{4} = \frac{1}{2}$

Likelihood

🎯 In this topic you will

🧠 Key Words

Using Probability to Make Decisions

Needing the Right Conditions

❓ EXERCISES

🧠 Think like a Mathematician

📘 What we've learned

Related Past Papers

Related Tutorials

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