Handling Data · Chance or Probability · Lesson 3

Calculating Probability as Fractions, Decimals, Percentages and Ratios

Probability puts a number on chance. In this lesson you will use the sample space to calculate the probability of an event and write it as a fraction, a decimal, a percentage and a ratio. You will use it to answer questions about dice, coins, spinners and bags of objects, just as a weather forecaster says 'a 30% chance of rain'.

NaCCA B7.4.2.1.3 ⏱ about 55 min

By the end you will be able to

✔State and use the formula P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
✔Calculate the probability of an event for a die, coin, spinner or bag of objects.
✔Express a probability as a fraction, decimal, percentage and ratio, and convert between them.
✔Use P(not A) = 1 − P(A) to solve problems.

Key words

Probability: A number from 0 to 1 that measures how likely an event is to happen.
Favourable outcome: An outcome that makes the event happen.
Equally likely: Outcomes that all have the same chance of happening, like the six faces of a fair die.
Complement (not A): The event that A does not happen. P(not A) = 1 − P(A).

Curriculum reference

Content standard: B7.4.2.1 Identify the sample space for a probability experiment involving single events and express the probabilities of given events as fractions, decimals, percentages and/or ratios to solve problems.

Indicator: Calculate the probability of the event and express the probability as fractions, decimals, percentages and/or ratios.

Starter

The weather forecast says 30%

On the radio the Ghana Meteorological Agency says: 'There is a 30% chance of rain in Kumasi this afternoon.' Should Yaw carry his umbrella? Thirty percent is less than half, so rain is unlikely but not out of the question. Probability gives a number to the words 'impossible', 'unlikely', 'likely' and 'certain', so that everyone can compare chances exactly.

1. The probability formula

When all outcomes of an experiment are equally likely, the probability of an event is the fraction of the outcomes that are favourable to it:

$P (event) = \frac{number of favourable outcomes}{total number of possible outcomes}$

For a fair die, the sample space is {1, 2, 3, 4, 5, 6}, so there are 6 equally likely outcomes. The event 'an even number' has favourable outcomes 2, 4 and 6, so $P (even) = \frac{3}{6} = \frac{1}{2}$ .

▶ Count the favourable sectors (the gold ones) and divide by the total number of sectors.

1List the sample space and count the total number of outcomes.
2List the favourable outcomes, those that make the event happen, and count them.
3Write the fraction favourable ÷ total, and simplify.
4Convert to a decimal, percentage or ratio if asked.

2. Fractions, decimals, percentages and ratios

A probability can be written in four equivalent ways. They all say the same thing, so choose the one that suits the question.

Four ways to write a probability
Form	How to get it	Example for P = 1/4
Fraction	favourable ÷ total, simplified	$\frac{1}{4}$
Decimal	divide the numerator by the denominator	0.25
Percentage	decimal × 100, then add %	25%
Ratio	favourable : unfavourable	1 : 3

Notice the ratio: it compares the number of favourable outcomes with the number of unfavourable outcomes. For rolling a 6 the ratio is 1 : 5 (one way to win and five ways to lose), although the fraction is $\frac{1}{6}$ . Do not confuse the two.

When a fraction is a recurring decimal such as $\frac{1}{3} = 0.3333...$ , we round to a sensible number of places, usually 2 or 3 decimal places, and the percentage becomes 33.3%.

3. Probability with a single die

Let us calculate several probabilities for one fair die. Each outcome 1 to 6 has a probability of $\frac{1}{6}$ . The table lists the favourable outcomes and writes each probability in all four ways.

Probabilities for one fair die (ratio = favourable : unfavourable)
Event: rolling…	Favourable outcomes	Fraction	Decimal	Percentage	Ratio
factors of 60	{1, 2, 3, 4, 5, 6}	1	1	100%	1 : 0
a multiple of 3	{3, 6}	1/3	0.333	33.3%	1 : 2
factors of 2	{1, 2}	1/3	0.333	33.3%	1 : 2
divisors of 12	{1, 2, 3, 4, 6}	5/6	0.833	83.3%	5 : 1
a 3 or greater	{3, 4, 5, 6}	2/3	0.667	66.7%	2 : 1
factors of 8	{1, 2, 4}	1/2	0.5	50%	1 : 1
factors of 6	{1, 2, 3, 6}	2/3	0.667	66.7%	2 : 1
divisors of 30	{1, 2, 3, 5, 6}	5/6	0.833	83.3%	5 : 1
a 3 or smaller	{1, 2, 3}	1/2	0.5	50%	1 : 1

Look at the first row: every number on a die is a factor of 60, so the event is certain. The probability is 1, which is 100%, and the ratio is 1 : 0 because there is no unfavourable outcome. Compare with 'factors of 2', where only 1 and 2 work, so the probability is $\frac{2}{6} = \frac{1}{3}$ . Row 9 shows 'a 3 or smaller', which is $\frac{3}{6} = \frac{1}{2} = 0.5 = 50%$ with ratio 1 : 1 — an even chance.

▶ The favourable part is 4 of the 6 equal parts, so the probability is 4/6 = 2/3.

4. Coins, bags, and the probability that an event does NOT happen

The same method works for any experiment with equally likely outcomes. A fair coin has 2 outcomes, so $P (head) = \frac{1}{2} = 0.5 = 50%$ . A bag with 5 red, 3 blue and 2 green beads has 10 beads, so $P (red) = \frac{5}{10} = \frac{1}{2}$ and $P (green) = \frac{2}{10} = \frac{1}{5} = 0.2$ .

If A is an event, 'not A' is the event that A does not happen. Together A and not A cover all outcomes, so their probabilities add to 1:

$P (not A) = 1 - P (A)$

For example, the probability of not rolling a 6 is $1 - \frac{1}{6} = \frac{5}{6}$ . This shortcut is helpful when counting the 'not' outcomes is harder than counting the outcomes of A.

We can also work backwards. If the probability of drawing a mango sweet from a bag of 25 sweets is 0.24, then the number of mango sweets is $0.24 \times 25 = 6$ . In general, number of favourable outcomes = probability × total number of outcomes.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

A coin and a die

Find (a) P(tail) when a fair coin is tossed, (b) P(5) when a fair die is thrown. Give (b) as a fraction and as a decimal to 3 decimal places.

Show the solution

Step 1(a) Outcomes: head, tail. P(tail) = 1/2 = 0.5.
Step 2(b) Outcomes: 1 to 6. P(5) = 1/6.
Step 3(b) 1 ÷ 6 = 0.1666… ≈ 0.167.

Answer: (a) 1/2 (0.5); (b) 1/6 ≈ 0.167

Example 2

Multiples of 3

A fair die is thrown. Find the probability of obtaining a multiple of 3, as a fraction, decimal, percentage and ratio.

Show the solution

Step 1Favourable outcomes: 3 and 6, so 2 out of 6.
Step 2Fraction = 2/6 = 1/3.
Step 3Decimal = 1 ÷ 3 = 0.333… ≈ 0.333; percentage ≈ 33.3%.
Step 4Ratio favourable : unfavourable = 2 : 4 = 1 : 2.

Answer: 1/3, 0.333, 33.3%, 1 : 2

Example 3

Divisors of 12

A fair die is thrown. Find the probability of obtaining a divisor of 12.

▶ Only the 5 is not a divisor of 12, so five of the six sectors are favourable.

Show the solution

Step 1Divisors of 12 are 1, 2, 3, 4, 6, 12. Those on a die: 1, 2, 3, 4, 6, which is 5 outcomes.
Step 2P = 5/6.
Step 3As a decimal: 5 ÷ 6 = 0.8333… ≈ 0.83; as a percentage ≈ 83.3%.

Answer: 5/6 ≈ 0.83 ≈ 83.3%

Example 4

Beads in a bag

A bag holds 5 red, 3 blue and 2 green beads. One is drawn at random. Find (a) P(red), (b) P(not green), (c) P(yellow).

Show the solution

Step 1Total beads = 5 + 3 + 2 = 10.
Step 2(a) P(red) = 5/10 = 1/2 = 0.5.
Step 3(b) P(not green) = 1 − 2/10 = 8/10 = 4/5 = 0.8.
Step 4(c) There are no yellow beads, so P(yellow) = 0/10 = 0 (impossible).

Answer: (a) 1/2, (b) 4/5, (c) 0

Example 5

Working backwards

The probability of picking a mango sweet from a bag of 25 sweets is 0.24. How many mango sweets are in the bag?

Show the solution

Step 1Number of mango sweets = probability × total.
Step 2= 0.24 × 25 = 6.
Step 3Check: 6/25 = 0.24 ✓

Answer: 6 mango sweets

Summary

P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.
Probability is a number from 0 (impossible) to 1 (certain), and can be written as a fraction, decimal, percentage or ratio.
Ratio = favourable : unfavourable, which is different from the fraction favourable / total.
P(not A) = 1 − P(A).
Number of favourable outcomes = probability × total outcomes.

The weather forecast says 30%

1. The probability formula

2. Fractions, decimals, percentages and ratios

3. Probability with a single die

4. Coins, bags, and the probability that an event does NOT happen

Worked examples

A coin and a die

Multiples of 3

Divisors of 12

Beads in a bag

Working backwards

Summary

Practice: 15 questions