Year 10 Maths Core Probability

Probability Basics

Q: How do you write a probability as a percentage?

Multiply the fraction or decimal by 100. For example 3 out of 8 is 0.375, which is 37.5 percent.

Q: How do you find the number of items from a probability?

Use the probability as parts of the whole. If 5 out of 8 parts are white and there are 25 white items, one part is 5 items and the whole is 40.

20 practice questions 1 video lesson Theory + worked examples

Probability Basics in NSW Year 10 Core maths revises how to find the probability of an event when every outcome is equally likely (NESA outcome MA4‑PRO‑C‑01, revising Stage 4). The probability is the number of favourable outcomes divided by the total number of outcomes, written as a fraction, a decimal or a percentage.

You will find probabilities for counters, marbles, dice, spinners, the letters of words like AUSTRALIA and numbered cards, including primes, multiples and two-digit numbers with matching digits. You will also work backwards from a probability to the number of marbles in a jar, find how many lollies to add to change a probability, and handle a raffle where the first winning ticket is not put back. It is part of Year 10 Core Probability.

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Probability Basics - Video - Probability, Sample Spaces, and the Complement Rule Watch

Theory

This Year 10 maths page revises probability from earlier years: list the sample space, count the favourable outcomes, and write \(P(\text{event})\) as a fraction, decimal or percentage. It also shows how to use a ratio, how to find a missing number of items from a probability, and how adding items changes a probability.

A chance experiment is something with an uncertain result, such as rolling a die. Each possible result is an outcome.

The sample space \(S\) is the list of all possible outcomes. For one roll of a die, \(S = \{1, 2, 3, 4, 5, 6\}\).

An event is a set of outcomes, such as "rolling an even number". The outcomes in the event are the favourable outcomes.

Outcomes are equally likely when each has the same chance, as with a fair die, a spinner with equal sectors, or choosing "at random".

The probability \(P(\text{event})\) is a number from \(0\) to \(1\). It can be written as a fraction, a decimal or a percentage.

The sectors are equal, so each of the \(8\) outcomes is equally likely. Three are red: \(P(\text{red}) = \dfrac{3}{8}\).

List the sample space, then count the favourable outcomes: \(P = \dfrac{2}{6} = \dfrac{1}{3}\).

For equally likely outcomes:

\[P(\text{event}) = \dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}\]

P (event) = \frac{favourable outcomes}{total outcomes}

Repeated letters. Count every card, including repeats. In BANANA there are \(6\) cards and \(3\) of them are A, so \(P(\text{A}) = \dfrac{3}{6} = \dfrac{1}{2}\).

Counting whole numbers. From \(25\) to \(60\) there are \(60 - 25 + 1 = 36\) numbers. Remember that \(1\) is not prime: the primes up to \(20\) are \(2, 3, 5, 7, 11, 13, 17, 19\).

Ratios. Red to blue \(2 : 7\) means \(9\) parts, so \(P(\text{red}) = \dfrac{2}{9}\).

Things that change. If an item is taken out and not put back, the total goes down by one. After one of \(300\) raffle tickets is drawn, a second draw is from \(299\) tickets. When only one position decides the event (such as the first digit of a number made from shuffled digits), just look at that position.

How to find a probability

Check that the outcomes are equally likely.
Count the total number of outcomes (list them if it helps).
Count the favourable outcomes. For "not" or "but not", cross out the ones that do not fit.
Divide and simplify. Change to a decimal or percentage if asked.

How to work backwards

Find the probability of the group you know the number of (use \(1 -\) the given probability if needed).
Use parts: the fraction tells you how many parts that group is; divide to find one part.
Or write an equation such as \(\dfrac{5 + k}{20 + k} = \dfrac{2}{5}\) and solve it. Check your answer.

Example 1 — Counting favourable outcomes

The numbers \(1\) to \(40\) are written on cards and one card is chosen at random. Find the probability that the number is:
(i) a multiple of \(6\)
(ii) a multiple of \(5\) but NOT a multiple of \(10\).

Solution

There are \(40\) equally likely outcomes.

(i) Multiples of \(6\): \(6, 12, 18, 24, 30, 36\). That is \(6\) cards.

\(P(\text{multiple of } 6)\)

\(=\)

\(\dfrac{6}{40} = \boldsymbol{\dfrac{3}{20}}\)

(ii) Multiples of \(5\): \(5, 10, 15, 20, 25, 30, 35, 40\). Cross out the multiples of \(10\), leaving \(5, 15, 25, 35\).

\(P\)

\(=\)

\(\dfrac{4}{40} = \boldsymbol{\dfrac{1}{10}}\)

P = \frac{1}{10}

Example 2 — Fraction, decimal and percentage

A bag holds \(6\) red, \(5\) green and \(9\) white counters. One counter is drawn at random. Find:
(i) \(P(\text{green})\)
(ii) \(P(\text{not white})\). Give each as a fraction, a decimal and a percentage.

Solution

Total: \(6 + 5 + 9 = 20\) counters.

\(P(\text{green})\)	\(=\)	\(\dfrac{5}{20} = \dfrac{1}{4}\)
\(\)	\(=\)	\(\boldsymbol{0.25 = 25\%}\)

(ii) "Not white" means red or green: \(6 + 5 = 11\) counters.

\(P(\text{not white})\)	\(=\)	\(\dfrac{11}{20}\)
\(\)	\(=\)	\(\boldsymbol{0.55 = 55\%}\)

P = \frac{11}{20} = 0.55

Example 3 — Working backwards to a total

A bag holds only blue and white beads. The probability of drawing a blue bead is \(\dfrac{3}{8}\). There are \(25\) white beads. How many beads are in the bag, and how many are blue?

Solution

The beads are blue or white, so \(P(\text{white}) = 1 - \dfrac{3}{8} = \dfrac{5}{8}\).

So \(5\) parts out of \(8\) are white, and those \(5\) parts are \(25\) beads.

\(1 \text{ part}\)	\(=\)	\(25 \div 5 = 5\)
\(\text{total}\)	\(=\)	\(8 \times 5 = \boldsymbol{40}\)
\(\text{blue}\)	\(=\)	\(3 \times 5 = \boldsymbol{15}\)

total = 40

Check: \(\dfrac{15}{40} = \dfrac{3}{8}\) ✓

Example 4 — Changing the probability

A jar holds \(5\) orange and \(15\) purple lollies. How many orange lollies must be added so that the probability of choosing an orange lolly is \(\dfrac{2}{5}\)?

Solution

Let \(k\) orange lollies be added. The orange count and the total both go up by \(k\); the purple count stays at \(15\).

\(\dfrac{5 + k}{20 + k}\)	\(=\)	\(\dfrac{2}{5}\)
\(5(5 + k)\)	\(=\)	\(2(20 + k)\)
\(25 + 5k\)	\(=\)	\(40 + 2k\)
\(3k\)	\(=\)	\(15\)
\(k\)	\(=\)	\(\boldsymbol{5}\)

k = 5

Check: \(\dfrac{10}{25} = \dfrac{2}{5}\) ✓. Add \(5\) orange lollies.

Common pitfalls

Use the total, not the other group. With \(4\) red and \(6\) blue, \(P(\text{red}) = \dfrac{4}{10}\), not \(\dfrac{4}{6}\).

1 is not a prime number. A prime has exactly two factors. \(1\) has only one, so leave it out when you count primes.

Adding items changes the total too. Adding \(3\) red to \(4\) red and \(6\) blue gives \(\dfrac{7}{13}\), not \(\dfrac{7}{10}\).

Simplify, then convert carefully. \(\dfrac{9}{12} = \dfrac{3}{4} = 0.75 = 75\%\). Round a percentage only at the end.

Frequently asked questions

How do you calculate a probability?

When the outcomes are equally likely, divide the number of favourable outcomes by the total number of outcomes, then simplify.

What is a sample space?

It is the list of every possible outcome of a chance experiment. For one coin toss, \(S = \{\text{H}, \text{T}\}\).

What does 'at random' mean?

It means every item has the same chance of being chosen, so the outcomes are equally likely.

How do you write a probability as a percentage?

Multiply the fraction or decimal by \(100\). For example \(\dfrac{3}{8} = 0.375 = 37.5\%\).

How do you find the number of items from a probability?

Use the probability as parts of the whole. If \(5\) out of \(8\) parts are white and there are \(25\) white items, one part is \(5\) items and the whole is \(40\).

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Probability Range

Probability Basics

📖 Theory

How to find a probability

How to work backwards

Common pitfalls

Frequently asked questions

Theory