Probability Tables and Two-Way Tables

This lesson focuses on reading and completing two-way tables, using them to find probabilities, and working with probability distribution tables. Two-way table questions are a staple of the Edexcel GCSE Mathematics (1MA1) exam and appear on both Foundation and Higher papers.

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What Is a Two-Way Table?

A two-way table displays data that has been classified by two different categories. The rows represent one category and the columns represent another. The table includes row totals, column totals and a grand total.

Example Structure

	Category A	Category B	Total
Group 1	a	b	a + b
Group 2	c	d	c + d
Total	a + c	b + d	a + b + c + d

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Reading Two-Way Tables

Worked Example 1

A survey asks 120 students whether they prefer football or rugby.

	Football	Rugby	Total
Boys	35	25	60
Girls	28	32	60
Total	63	57	120

(a) How many girls prefer rugby? (b) How many students prefer football? (c) What fraction of the boys prefer football?

Solution:

(a) Read from the Girls row and Rugby column: 32

(b) Read the Football column total: 63

(c) Boys who prefer football = 35, total boys = 60. Fraction = 35/60 = 7/12

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Completing Two-Way Tables

Edexcel frequently gives you a two-way table with missing values. You need to use the relationship:

Row total = sum of entries in that row Column total = sum of entries in that column Grand total = sum of all row totals = sum of all column totals

Worked Example 2

Complete the two-way table.

	Walk	Bus	Car	Total
Year 7	15	?	10	45
Year 8	?	16	?	55
Total	35	?	?	100

Solution:

Step 1: Year 7 Bus = 45 − 15 − 10 = 20

Step 2: Year 8 Walk = 35 − 15 = 20

Step 3: Year 8 Car = 55 − 20 − 16 = 19

Step 4: Total Bus = 20 + 16 = 36

Step 5: Total Car = 10 + 19 = 29

Check: 35 + 36 + 29 = 100 ✓

Completed table:

	Walk	Bus	Car	Total
Year 7	15	20	10	45
Year 8	20	16	19	55
Total	35	36	29	100

Edexcel Exam Tip: Always check that your completed rows and columns add up to the correct totals. This is easy marks and catches careless errors.

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Finding Probabilities from Two-Way Tables

Once you have a completed two-way table, you can find probabilities by dividing the relevant frequency by the appropriate total.

Worked Example 3

Using the table above, a student is chosen at random. Find:

(a) P(the student walks to school) (b) P(the student is in Year 8 and takes the bus) (c) P(the student takes the car, given that they are in Year 7)

Solution:

(a) Total who walk = 35, grand total = 100. P(walk) = 35/100 = 7/20

(b) Year 8 and bus = 16, grand total = 100. P(Year 8 and bus) = 16/100 = 4/25

(c) This is a conditional probability. We are told the student is in Year 7 (total = 45). Year 7 car = 10. P(car | Year 7) = 10/45 = 2/9

Key Point: For conditional probability, the denominator changes to the row or column total of the given condition, not the grand total.

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Probability Distribution Tables

A probability distribution table lists all possible outcomes of an event alongside their probabilities. The probabilities must sum to 1.

Worked Example 4

A biased spinner has the following probability distribution:

Score	1	2	3	4	5
Probability	0.1	0.15	0.3	0.25	?

(a) Find the probability of scoring 5. (b) The spinner is spun 200 times. How many times would you expect a score of 3?

Solution:

(a) P(5) = 1 − (0.1 + 0.15 + 0.3 + 0.25) = 1 − 0.8 = 0.2

(b) Expected number of 3s = 0.3 × 200 = 60

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Harder Two-Way Table Problems

Worked Example 5

240 students are surveyed about their lunch choice.

• 100 students are in Year 10; the rest are in Year 11.
• 60% of Year 10 students choose a hot meal.
• 45 Year 11 students choose a cold meal.
• The rest choose a hot meal.

Complete the table and find the probability that a randomly chosen student has a cold meal.

Solution:

Year 11 students = 240 − 100 = 140 Year 10 hot = 60% of 100 = 60 Year 10 cold = 100 − 60 = 40 Year 11 cold = 45 Year 11 hot = 140 − 45 = 95

	Hot	Cold	Total
Year 10	60	40	100
Year 11	95	45	140
Total	155	85	240

P(cold meal) = 85/240 = 17/48

Worked Example 6

A club of 160 members is classified by age group and whether they play tennis or squash.

	Tennis	Squash	Total
Under 18	25	?	40
18 or over	?	50	?
Total	?	?	160

(a) Complete the table. (b) A member is chosen at random. Find P(plays squash). (c) Find P(Under 18 ∩ Tennis) using set notation.

Solution:

(a)

• Under 18 Squash = 40 − 25 = 15
• 18 or over total = 160 − 40 = 120
• 18 or over Tennis = 120 − 50 = 70
• Total Tennis = 25 + 70 = 95
• Total Squash = 15 + 50 = 65

Check: 95 + 65 = 160 ✓

(b) P(squash) = 65/160 = 13/32

(c) P(Under 18 ∩ Tennis) = 25/160 = 5/32

Worked Example 7

The probability distribution for a biased five-sided spinner is:

Score	1	2	3	4	5
Probability	0.12	2x	0.18	3x	0.20

(a) Find the value of x. (b) Find P(score ≥ 3). (c) The spinner is spun 400 times. How many times would you expect a score of 4?

Solution:

(a) All probabilities must sum to 1: 0.12 + 2x + 0.18 + 3x + 0.20 = 1 5x + 0.50 = 1 5x = 0.50 x = 0.10

(b) P(3) = 0.18, P(4) = 3(0.10) = 0.30, P(5) = 0.20. P(score ≥ 3) = 0.18 + 0.30 + 0.20 = 0.68

(c) Expected 4s = 0.30 × 400 = 120

Worked Example 8

200 adults are surveyed about whether they drive to work and whether they live in the city.