Probability from Tables and Charts

Read a probability from a frequency table or chart, find x when a table of probabilities adds to 1, and turn a probability into a number of times or a total.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 48 real past-paper questions below

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What a Probability Table Shows

Exam tables come in two kinds. Look at the numbers inside before you start.

The table holds	You do
counts: how many	count over the total
probabilities: 0.3, 0.25	they add up to 1

A chart works the same way. Read the bars or the slices, then use them like a table.

Rule

A count table: add every row to get the total first. A probability table: every row together makes 1.

Reading a Frequency Table

The 1 or 2-mark question: one person or thing is chosen at random. Find the probability from the table.

60 flights were late. The table shows how many minutes late.

Minutes late	Flights
$0 < t \leq 5$	14
$5 < t \leq 10$	22
$10 < t \leq 20$	15
$20 < t \leq 30$	9

Worked example

1
$14 + 22 + 15 + 9 = 60$ — Add every row. This is the total number of flights, and it goes on the bottom of each probability.
2
$P (t \leq 5) = \frac{14}{60}$ — Only the first row is 5 minutes or less, so 14 flights out of 60.
3
$= \frac{7}{30}$ — Divide the top and the bottom by 2 to simplify.
4
$P (t > 10) = \frac{15 + 9}{60}$ — More than 10 minutes means the last two rows, so add their counts.
5
$= \frac{24}{60} = \frac{2}{5}$ — 15 + 9 = 24. Then divide the top and the bottom by 12.

Common mistake

Dividing by 4 because there are 4 rows. The bottom is the number of flights, 60.

From a pie chart. A pie chart shows how students travel to school: bus 150°, walk 96°, car 114°. One student is chosen at random. Find P(walk).

Worked example

1
$150 + 96 + 114 = 360$ — The slices make the whole circle, 360°. The circle stands for everyone, so 360 plays the part of the total.
2
$P (walk) = \frac{96}{360}$ — The walk slice's angle over the whole circle.
3
$= \frac{4}{15}$ — Divide the top and the bottom by 24 to simplify.

Key idea

A bar chart? Read each bar's height, then use one bar over all the bars added, just like a table.

Find x in a Probability Table

The 4-mark question: a biased dice. Some probabilities are written with x. Make them add to 1, solve for x, then answer the question.

Number	Probability
1	$2 x$
2	0.14
3	$x$
4	$3 x$
5	0.22
6	$2 x$

The dice is thrown 250 times. Estimate how many times it lands on an even number.

Worked example

1
$2 x + x + 3 x + 2 x = 8 x$ — Collect the rows with x in them: 2 + 1 + 3 + 2 makes 8 lots of x.
2
$0.14 + 0.22 = 0.36$ — Add the rows that are plain numbers.
3
$8 x + 0.36 = 1$ — The whole table must add up to 1, so write that as an equation.
4
$8 x = 1 - 0.36 = 0.64$ — Take 0.36 from both sides.
5
$x = 0.64 \div 8 = 0.08$ — Divide both sides by 8 to find x.
6
$P (even) = 0.14 + 3 x + 2 x$ — Even means 2, 4 and 6, so add those three rows.
7
$= 0.14 + 3 (0.08) + 2 (0.08)$ — Put x = 0.08 into each row.
8
$= 0.14 + 0.24 + 0.16 = 0.54$ — Multiply first: 3 × 0.08 = 0.24 and 2 × 0.08 = 0.16. Then add.
9
$0.54 \times 250 = 135$ — Probability times throws: about 135 even numbers in 250 throws.

Common mistake

Using x for row 4. Row 4 is 3x, so it is $3 \times 0.08 = 0.24$ . Multiply every x before you add.

Key idea

Given a ratio instead, like blue : green = 5 : 2? Call them 5x and 2x, then use the same method. Writing “8x + 0.36 = 1” earns a method mark on its own.

Key idea

"Show that x + y = 0.19"? Write the sum of the whole row equal to 1, then take the known numbers across.

From One Count to the Total

The 4-mark twist: the table has probabilities, and one count is given. Use it to find how many there are in total.

Members chose their favourite book: comedy 0.2, romance 0.35, mystery $2 x$ , thriller $x$ . 36 members chose comedy. How many chose thriller?

Worked example

1
$0.2 + 0.35 = 0.55$ — Add the probabilities you know: comedy and romance.
2
$3 x = 1 - 0.55 = 0.45$ — Mystery 2x and thriller x make 3x. They share what is left of 1.
3
$x = 0.45 \div 3 = 0.15$ — Divide by 3. This is P(thriller).
4
$0.2 \times total = 36$ — The 36 comedy members are 0.2 of everyone, so probability times total gives 36.
5
$total = 36 \div 0.2 = 180$ — Divide both sides by 0.2 to find all the members.
6
$0.15 \times 180 = 27$ — Probability times total: 27 members chose thriller.

Formula

total = count \div P

Divide the count by its probability to get the total.

Common mistake

Working out $36 \times 0.2$ . The 36 is part of the total, so the total must be bigger: divide.

Counts Written with x

Some questions write the counts in a bag with x, give the total, and give one probability. Turn the probability into an equation.

A bag has 60 counters. There are x + 5 red, 2x − 3 blue and x green. The rest are yellow. P(red) = $\frac{1}{4}$ . Find P(yellow).

Worked example

1
$\frac{x + 5}{60} = \frac{1}{4}$ — Red is x + 5 counters out of 60. Set that equal to the probability you are given.
2
$x + 5 = 60 \times \frac{1}{4} = 15$ — Multiply both sides by 60. A quarter of 60 is 15, so there are 15 red counters.
3
$x = 10$ — Take 5 from both sides.
4
$blue = 2 (10) - 3 = 17$ — Put x = 10 into blue's expression: 20 − 3 = 17.
5
$green = 10$ — Green is x counters, so 10.
6
$yellow = 60 - 15 - 17 - 10 = 18$ — The rest of the bag is yellow, so take the other colours from 60.
7
$P (yellow) = \frac{18}{60} = \frac{3}{10}$ — Yellow counters over the total, then divide the top and the bottom by 6.

Common mistake

Stopping at x = 10. x is not a count of any colour you were asked about. Put it back into each expression first.

Key idea

Check: 15 + 17 + 10 + 18 = 60, the whole bag.

Which Method? Read the Question

Look at what is inside the table, then at what the question wants.

You see	You do
A table of counts	cell or rows over the total
A pie chart	slice angle over 360
Probabilities with x	add to 1, solve for x
Estimate the number of times	probability × trials
A count is given too	count ÷ probability = total
Counts written with x, and a probability	count over total = probability, solve
Only the girls, given that	Conditional Probability
Experiment results	Relative Frequency and Expectation
Two facts about each person	Combined Events with Two-Way Tables

Key idea

Check at the end: every probability is between 0 and 1, and a count is a whole number.

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