Probability (Grade 9) — Mathematics Grades 8–11

Finding the probability of a compound event using a two-way table or a tree diagram.

Gr 9 Term 4 — ProbabilityTerm 4

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What gets asked

Find the probability of a compound event using a two-way table.
Use a tree diagram to find the probability of a compound event.
Predict how many times an outcome should happen in a number of trials.
Compare a relative frequency from an experiment with a theoretical probability.

You must be able to

List every possible outcome of two combined events without missing any.
Multiply along the branches of a tree diagram to find a combined probability.
Multiply a probability by the number of trials to predict a frequency.
Explain why an experiment's results can differ from the theoretical probability.

Traps that cost marks

Missing outcomes by not branching every existing branch at the next stage. Every branch at one stage must split again at the next stage. Check none was skipped.
Treating two different orders, such as red-then-blue and blue-then-red, as the same outcome. Order matters: red-then-blue and blue-then-red are two separate outcomes.
Expecting a predicted frequency to happen exactly in a real experiment. A prediction is an expected value. The real result can be close but different.

Worked example

A bag has 3 red balls and 2 blue balls. Zanele draws one ball, replaces it, then draws again. Calculate P(red then blue).

P(red) $= \frac{3}{5}$ and P(blue) $= \frac{2}{5}$ .
The ball is replaced, so multiply the two probabilities.
P(red then blue) $= \frac{3}{5} \times \frac{2}{5} = \frac{6}{25}$ .

Probability and frequency

How likely something is, how often it should turn up, and why the real count seldom matches.

outcome: One of the things that can happen. Drawing a red bead 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 between $0$ and $1$ .
relative frequency: How often the outcome really happened, divided by the number of trials.
compound event: An event made of two or more stages, such as tossing a coin twice.
tree diagram: A branching picture showing every outcome of a compound event.

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

To predict how often something will happen, multiply the probability by the number of trials. Toss a fair coin $50$ times and you expect $25$ heads. You are predicting the heads, not the tosses.

The relative frequency is what really happened. It will usually differ a little from the probability, and that does not make the coin unfair. The next toss does not owe you anything either.

Two stages together make a compound event. List every outcome. Tossing two coins gives HH, HT, TH and TT: four outcomes, not three. HT and TH differ, because the order differs.

A tree diagram shows this clearly. Every branch at the first stage must split again at the second. Multiply the branches at each stage to get the total, and never add them.

Rules to remember

P(outcome) $= \frac{ways it happens}{all outcomes}$
expected number $=$ probability $\times$ number of trials

Examples

A bag holds

4

red beads and

6

blue ones. A bead is drawn and put back,

50

times over. How many reds are expected?

Worked answer

There are $4 + 6 = 10$ beads in the bag.
P(red) $= \frac{4}{10} = \frac{2}{5}$ .
Expected reds: $\frac{2}{5} \times 50 = 20$ .
So about $20$ of the $50$ draws should be red.

Answer: about $20$ reds

The question asked how many REDS, not how many draws there were.

That same bag is drawn from

50

times and red comes up

17

times. Is the bag unfair?

Worked answer

Relative frequency of red: $\frac{17}{50} = 0, 34$ .
The probability was $\frac{2}{5} = 0, 4$ .
So $0, 34$ sits close to $0, 4$ without matching it.
A gap that size is ordinary in $50$ trials, so the bag looks fine.

Answer: no, a difference that small is normal

Relative frequency wanders around the probability; it need not land on it.

A coin is tossed twice. Find the probability of getting two heads.

Worked answer

First toss: H or T. Each of those splits again on the second toss.
The outcomes are HH, HT, TH and TT, which is $2 \times 2 = 4$ .
Only one of those four is HH.
P(two heads) $= \frac{1}{4}$ .

Answer: $\frac{1}{4}$

Looking at one toss only would give $\frac{1}{2}$ , which is twice too big.

Traps

Expecting a predicted frequency to happen exactly in a real experiment. A prediction is an expected value. The real result can be close but different.
Calling a coin unfair as soon as the results differ from the probability. Small differences are normal. Only a big gap over many trials is evidence.
Treating two different orders, such as red-then-blue and blue-then-red, as the same outcome. Order matters: red-then-blue and blue-then-red are two separate outcomes.
Missing outcomes by not branching every existing branch at the next stage. Every branch at one stage must split again at the next stage. Check none was skipped.

Learn and practise “Probability and frequency” in the app →

Grade 9 Probability

What gets asked

You must be able to

Traps that cost marks

Worked example

Probability and frequency

Rules to remember

Examples

Traps

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