Probability (Grade 8) — Mathematics Grades 8–11

Listing outcomes, finding probabilities, and comparing them with relative frequency.

Gr 8 Term 4 — ProbabilityTerm 4

Practise Probability in the app →

What gets asked

List all the possible outcomes of a simple situation.
Write the probability of an outcome as a fraction.
Predict the relative frequency of an outcome over many trials.
Explain why relative frequency and probability are not always exactly equal.

You must be able to

List every outcome once, without missing or repeating any.
Put the number of favourable outcomes over the total number of outcomes.
Scale a probability up to predict results over many trials.
Explain a gap between relative frequency and probability using sample size.

Traps that cost marks

Treating identical-looking items in a bag as a single outcome. Each item is a separate outcome, even if two look the same.
Writing a probability as favourable outcomes over the outcomes that did not happen. A probability is always favourable outcomes over the total number of outcomes.
Predicting that every outcome will happen exactly the same number of times. Scale each outcome by its own probability. A more likely outcome should happen more often.
Deciding a coin or die is unfair as soon as one result differs from the expected probability. A small number of trials can differ from the probability just by chance. That does not mean it is unfair.

Worked example

A bag has 3 red, 4 blue and 5 green sweets. Find the probability of picking a blue sweet.

Total sweets $= 3 + 4 + 5 = 12$ .
Blue sweets $= 4$ .
$P (blue) = \frac{4}{12} = \frac{1}{3}$ .

Outcomes and relative frequency

Listing everything that can happen, saying how likely each one is, and checking that against a real try.

outcome: One of the things that can happen. Drawing a red sweet is one outcome.
trial: One go: one toss of a coin, or one draw from a bag.
probability: How likely an outcome is, written as a fraction from $0$ to $1$ .
equally likely: Every outcome has the same chance, as on a fair coin or a fair die.
relative frequency: How many times the outcome really happened, divided by the number of trials.

Start by listing every outcome, one by one. A bag holding $3$ red and $4$ blue sweets has $7$ outcomes, not $2$ . Two sweets of one colour are still two separate outcomes.

Probability counts ways. Put the number of ways your outcome can happen over the total number of outcomes. It runs from $0$ , meaning impossible, up to $1$ , meaning certain.

This year every situation has outcomes that are equally likely. A fair coin, a fair die and a well-mixed bag all behave that way, so counting the items is enough.

To predict, multiply the probability by the number of trials. A probability of $\frac{1}{3}$ over $60$ trials predicts about $20$ . You are predicting a number of draws, not a fraction.

The relative frequency is what really happened. It seldom lands exactly on the probability, and a small gap proves nothing about fairness. The more trials you do, the closer it usually comes.

Rules to remember

P(outcome) $= \frac{ways it can happen}{total outcomes}$
predicted number $=$ probability $\times$ number of trials

Examples

A bag holds

3

red,

4

blue and

5

green sweets. Find the probability of drawing a blue one.

Worked answer

Count every sweet: $3 + 4 + 5 = 12$ .
Each sweet is one outcome, even two greens that look alike.
Four of those twelve outcomes are blue.
P(blue) $= \frac{4}{12} = \frac{1}{3}$ .

Answer: $\frac{1}{3}$

The total goes underneath. Blue over the other eight sweets would be wrong.

A sweet is drawn from that bag and put back,

60

times over. How many blues should you expect?

Worked answer

The probability of blue is $\frac{1}{3}$ .
Multiply it by the number of trials: $\frac{1}{3} \times 60 = 20$ .
So about $20$ of the $60$ draws should be blue.

Answer: about $20$ blues

Red is less likely than blue, so the three colours will not come up equally often.

In those

60

draws blue actually came up

17

times. Is the bag unfair?

Worked answer

Relative frequency of blue: $\frac{17}{60}$ .
The probability was $\frac{1}{3} = \frac{20}{60}$ .
You expected $20$ and got $17$ , so $20 - 17 = 3$ .
A gap of $3$ in $60$ trials turns up easily by chance.

Answer: no, a gap that small is ordinary

Doing $600$ draws instead would usually bring the relative frequency nearer.

Traps

Treating identical-looking items in a bag as a single outcome. Each item is a separate outcome, even if two look the same.
Writing a probability as favourable outcomes over the outcomes that did not happen. A probability is always favourable outcomes over the total number of outcomes.
Predicting that every outcome will happen exactly the same number of times. Scale each outcome by its own probability. A more likely outcome should happen more often.
Deciding a coin or die is unfair as soon as one result differs from the expected probability. A small number of trials can differ from the probability just by chance. That does not mean it is unfair.

Learn and practise “Outcomes and relative frequency” in the app →

Grade 8 Probability

What gets asked

You must be able to

Traps that cost marks

Worked example

Outcomes and relative frequency

Rules to remember

Examples

Traps

About this material