Year 10 Maths Core Probability

Probability Range

Q: What is the range of a probability?

A probability is always from 0 to 1. 0 means impossible, 1 means certain, and 0.5 is an even chance.

Q: Can a probability be negative or bigger than 1?

No. A probability below 0 or above 1 (or above 100 percent) is always a mistake.

Q: How do you find a missing probability?

Add the probabilities you know and subtract the total from 1. For example, if win is 0.55 and draw is 0.15, lose is 0.3.

20 practice questions 1 video lesson Theory + worked examples

Probability Range in NSW Year 10 Core maths is about the scale from 0 to 1 (NESA outcome MA4‑PRO‑C‑01, revising Stage 4). An impossible event has probability 0, a certain event has probability 1, an even chance is \(0.5\), and the probabilities of all the possible outcomes of an experiment add to 1.

You will decide which numbers can be probabilities, describe likelihood in words, and find a missing probability when the others are known, using fractions, decimals and percentages. You will also work with a weighted die, a spinner with one sector twice the size of the others and a spinner where one colour is twice as likely as another, and explain why a team cannot have a \(0.7\) chance of winning and a \(0.4\) chance of losing. It is part of Year 10 Core Probability.

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Probability Range - Video - Introduction to Probability Watch

Theory

This Year 10 maths page revises the earlier probability scale: every probability lies from \(0\) (impossible) to \(1\) (certain), and the probabilities of all the outcomes of an experiment add to \(1\). It shows how to find a missing probability, turn it into a number of items, solve an equation such as \(x + 3x + 0.2 = 1\), and handle biased dice and unequal spinners.

Every probability is a number from \(0\) to \(1\): \(0 \le P(\text{event}) \le 1\). As a percentage, it is from \(0\%\) to \(100\%\).

A probability of \(0\) means the event is impossible. A probability of \(1\) means it is certain. \(0.5\) is an even chance; below \(0.5\) is unlikely and above \(0.5\) is likely.

The probabilities of all the outcomes of an experiment add to \(1\), provided the outcomes cover every possibility and no two can happen at once (such as win, draw and lose).

A biased (weighted) die or coin does not have equally likely outcomes. Its probabilities still add to \(1\).

On a spinner with unequal sectors, each probability is the sector's fraction of the whole circle.

Every probability sits on this scale. \(0\) means it cannot happen and \(1\) means it must happen.

When the outcomes cover every possibility and cannot happen together, their probabilities add to \(1\).

The two facts used on this page:

\[0 \le P(E) \le 1 \qquad P(E_1) + P(E_2) + \dots + P(E_n) = 1\]

0 \leq P (E) \leq 1

Percentages. They add to \(100\%\). If four outcomes are \(12\%\), \(30\%\), \(18\%\) and \(25\%\), the fifth is \(100\% - 85\% = 15\%\).

A biased die. If each of \(1\) to \(5\) has probability \(0.12\), then \(P(6) = 1 - 5 \times 0.12 = 0.4\).

"Twice as likely". If \(P(\text{blue})\) is twice \(P(\text{green})\), write them as \(2g\) and \(g\). With \(P(\text{red}) = 0.25\): \(3g = 0.75\), so \(g = 0.25\).

Sector areas. If one sector is three times the size of each of two equal sectors, count it as \(3\) parts: \(5\) parts in all, so its probability is \(\dfrac{3}{5}\).

Probabilities to numbers. Number of items \(=\) probability \(\times\) total. Total \(=\) number of items \(\div\) probability.

How to find a missing probability

Check that the listed outcomes are all the outcomes and cannot happen together.
Add the probabilities you know (use a common denominator for fractions).
Subtract from \(1\) (or from \(100\%\)).
Check that the answer is between \(0\) and \(1\).

How to solve for an unknown probability

Write every probability in terms of the pronumeral.
Set the sum equal to \(1\) and solve.
Substitute back to find each probability, and check they add to \(1\).

Example 1 — Could it be a probability?

Which of these could be a probability? \(0.85,\ \dfrac{7}{5},\ -0.2,\ 100\%,\ 0\). Describe each one that could, in words.

Solution

A probability must be from \(0\) to \(1\) (or \(0\%\) to \(100\%\)).

\(0.85\)	\(\)	\(\text{yes: likely}\)
\(\dfrac{7}{5} = 1.4\)	\(\)	\(\text{no: more than } 1\)
\(-0.2\)	\(\)	\(\text{no: less than } 0\)
\(100\% = 1\)	\(\)	\(\text{yes: certain}\)
\(0\)	\(\)	\(\text{yes: impossible}\)

The possible probabilities are \(0.85\), \(100\%\) and \(0\).

Example 2 — The missing probability

A bag holds only green, yellow and purple counters. \(P(\text{green}) = \dfrac{3}{10}\) and \(P(\text{yellow}) = \dfrac{1}{4}\).
(i) Find \(P(\text{purple})\).
(ii) There are \(60\) counters. How many are purple?

Solution

(i) The three probabilities add to \(1\). Use the common denominator \(20\).

\(P(\text{purple})\)	\(=\)	\(1 - \dfrac{3}{10} - \dfrac{1}{4}\)
\(\)	\(=\)	\(\dfrac{20}{20} - \dfrac{6}{20} - \dfrac{5}{20}\)
\(\)	\(=\)	\(\boldsymbol{\dfrac{9}{20}}\)

P = \frac{9}{20}

(ii) Multiply the probability by the total:

\(\dfrac{9}{20} \times 60\)

\(=\)

\(\boldsymbol{27}\)

There are \(27\) purple counters.

Example 3 — An equation from probabilities

A spinner has four colours with \(P(\text{A}) = x\), \(P(\text{B}) = x + 0.1\), \(P(\text{C}) = 2x\) and \(P(\text{D}) = 0.3\). Find \(x\), then \(P(\text{B})\) and \(P(\text{C})\).

Solution

The four colours are the only outcomes, so their probabilities add to \(1\).

\(x + (x + 0.1) + 2x + 0.3\)	\(=\)	\(1\)
\(4x + 0.4\)	\(=\)	\(1\)
\(4x\)	\(=\)	\(0.6\)
\(x\)	\(=\)	\(\boldsymbol{0.15}\)

\(P(\text{B})\)	\(=\)	\(0.15 + 0.1 = \boldsymbol{0.25}\)
\(P(\text{C})\)	\(=\)	\(2 \times 0.15 = \boldsymbol{0.3}\)

x = 0.15

Check: \(0.15 + 0.25 + 0.3 + 0.3 = 1\) ✓

Example 4 — Sectors that are not equal

A spinner has a red sector of \(90^\circ\), a blue sector of \(120^\circ\) and a green sector of \(150^\circ\). Find the probability of each colour.

Solution

The sectors are not equal, so each probability is the sector's share of the full turn, \(360^\circ\).

\(P(\text{red})\)	\(=\)	\(\dfrac{90}{360} = \boldsymbol{\dfrac{1}{4}}\)
\(P(\text{blue})\)	\(=\)	\(\dfrac{120}{360} = \boldsymbol{\dfrac{1}{3}}\)
\(P(\text{green})\)	\(=\)	\(\dfrac{150}{360} = \boldsymbol{\dfrac{5}{12}}\)

P (green) = \frac{5}{12}

Check: \(\dfrac{3}{12} + \dfrac{4}{12} + \dfrac{5}{12} = 1\) ✓

Common pitfalls

No probability is more than 1. \(1.3\), \(\dfrac{5}{4}\) and \(120\%\) cannot be probabilities. Negative numbers cannot be either.

Check the total. Win \(0.7\) and lose \(0.4\) cannot both be right: they add to \(1.1\), which is more than \(1\).

Unequal sectors are not equally likely. A spinner with \(3\) colours is not \(\dfrac{1}{3}\) each unless the sectors are equal.

Use a common denominator. \(1 - \dfrac{1}{5} - \dfrac{1}{2} = \dfrac{10 - 2 - 5}{10} = \dfrac{3}{10}\), not \(1 - \dfrac{2}{7}\).

Frequently asked questions

What is the range of a probability?

A probability is always from \(0\) to \(1\). \(0\) means impossible, \(1\) means certain, and \(0.5\) is an even chance.

Why do the probabilities of all outcomes add to 1?

One of the outcomes must happen, so together they are certain. A certain event has probability \(1\).

Can a probability be negative or bigger than 1?

No. A probability below \(0\) or above \(1\) (or above \(100\%\)) is always a mistake.

How do you find a missing probability?

Add the probabilities you know and subtract the total from \(1\). For example, if win is \(0.55\) and draw is \(0.15\), lose is \(0.3\).

What does a biased die mean?

Its faces are not equally likely, so some numbers come up more often. The probabilities of its six faces still add to \(1\).

Probability Range

📖 Theory

How to find a missing probability

How to solve for an unknown probability

Common pitfalls

Frequently asked questions

Theory