# Chance and Probability

> Describe chance in words and as a fraction, find probabilities with bags, spinners and dice, work backwards from a probability, and see what changes when items are not put back.

From Selective School Maths on WhizzNotes, free study notes for NSW students: https://whizznotes.com/selective-school-maths/probability/. Updated 4 November 2026.
Year 11 and Year 12 boxes go deeper than the Year 10 text around them.

### Story time: The lucky dip
At the school fair, the lucky dip bag holds 20 prizes: 12 pencils, 5 rubbers and 3 toy cars. Kai really wants a car.

"What are my chances?" he asks. His teacher says, "There are 3 cars out of 20 prizes, so the chance is 3 out of 20. Not likely, but possible."

Kai's friend pulls out a pencil and keeps it. Now there are 19 prizes and still 3 cars. "My chances just got a tiny bit better," says Kai.

**In short:** Probability is how likely something is, from 0 (impossible) to 1 (certain). When every outcome is equally likely, the probability is the number of ways it can happen over the total number of outcomes. If an item is taken out and not put back, the total goes down for the next pick.

## Before you start

- Simplifying fractions (see Types of Fractions and Equivalent Fractions).
- Fractions, decimals and percentages (see Fractions, Decimals and Percentages).

## Words for chance

| Word | Probability |
|---|---|
| Impossible | 0 |
| Unlikely | between 0 and $\frac{1}{2}$ |
| Even chance | $\frac{1}{2}$ |
| Likely | between $\frac{1}{2}$ and 1 |
| Certain | 1 |

## Probability as a fraction

### Key idea: Equally likely outcomes
$$\text{probability} = \frac{\text{number of ways it can happen}}{\text{total number of outcomes}}$$

### Worked example: The lucky dip
Probability of a toy car $= \frac{3}{20}$. Probability of a pencil $= \frac{12}{20} = \frac{3}{5}$.

Probability of **not** a pencil $= 1 - \frac{3}{5} = \frac{2}{5}$.

### Exam tip: The probabilities add to 1
All the outcomes together are certain: $\frac{12}{20} + \frac{5}{20} + \frac{3}{20} = 1$.

## Spinners and dice

On a fair dice, each number from 1 to 6 has a probability of $\frac{1}{6}$. The chance of an even number is $\frac{3}{6} = \frac{1}{2}$.

A spinner has 8 equal sections. If 3 of them show a number bigger than 0.5, the chance of landing on one is $\frac{3}{8}$. Only count sections, not colours or sizes, when the sections are equal.

## Working backwards

### Worked example: Finding the missing colour
A bag has 40 marbles: 14 green, some red and the rest yellow. The chance of red is $\frac{1}{2}$, so there are 20 red. Yellow $= 40 - 14 - 20 = 6$.

### Worked example: Ratios in a bag
A bag has three times as many green counters as white, and the same number of white as black. Say white is 1 part. Then black is 1 part and green is 3 parts: 5 parts in all. The chance of green is $\frac{3}{5}$.

## Not putting it back

### Key idea: Without replacement
After an item is taken out and kept, there is one fewer item in the bag, and one fewer of its kind.

### Worked example: Two out, then one more
A bag has 9 red and 5 blue marbles. Two are taken out and kept, and both are red. Now there are 7 red and 5 blue, 12 in all. The chance the next is blue is $\frac{5}{12}$.

### Challenge: Selective level: the missing fraction and two dice
A box has red, green and blue counters. The chance of red is $\frac{1}{4}$ and of green is $\frac{1}{3}$. There are 10 blue counters. The chance of blue is $1 - \frac{3}{12} - \frac{4}{12} = \frac{5}{12}$. So $\frac{5}{12}$ of the box is 10 counters, $\frac{1}{12}$ is 2, and the box holds 24.

Roll two dice and add them. There are $6 \times 6 = 36$ equally likely pairs. A total of 7 comes from (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) and (6, 1): 6 ways. The chance is $\frac{6}{36} = \frac{1}{6}$, the most likely total of all.

**Mind map**

```text
Probability
  Scale
    0 impossible
    1/2 even chance
    1 certain
  Fraction
    Ways over total
    All add to 1
    Not = 1 minus
  Backwards
    From probability to count
    Use parts for ratios
  Without replacement
    Total goes down
    One fewer of that kind
```

## Quick summary

- Probability goes from 0 (impossible) to 1 (certain).
- Probability $=$ ways it can happen $\div$ total outcomes.
- The probabilities of all outcomes add to 1, so P(not A) $= 1 -$ P(A).
- Work backwards: probability $\times$ total gives how many.
- Without replacement, the total and that kind each go down by 1 for the next pick.

## Quiz

Try these questions from the chapter quiz.

1. A bag has 3 red and 5 green balls. What is the probability of picking red?
   - A. $\frac{3}{5}$
   - B. $\frac{3}{8}$
   - C. $\frac{5}{8}$
   - D. $\frac{1}{3}$
   - Answer: B. $\frac{3}{8}$. 3 red out of 8 balls.

2. What is the probability of rolling a 7 on a normal dice?
   - A. 0
   - B. $\frac{1}{7}$
   - C. $\frac{1}{6}$
   - D. 1
   - Answer: A. 0. A dice only has 1 to 6, so it is impossible.

3. What is the probability of rolling an even number on a dice?
   - A. $\frac{1}{6}$
   - B. $\frac{1}{3}$
   - C. $\frac{1}{2}$
   - D. $\frac{2}{3}$
   - Answer: C. $\frac{1}{2}$. 2, 4 and 6: three out of six.

The whole chapter quiz is on the page, for signed-in students: https://whizznotes.com/selective-school-maths/probability/#quiz