Lesson Worksheet: Conditional Probability: Two-Way Tables Mathematics

In this worksheet, we will practice dealing with the concept of conditional probability using joint frequencies presented in two-way tables.

Q1:

The two-way table shows the ages and activity choices of a group of participants at a summer camp.

SwimmingClimbingRappelling
14 and Under15248
Over 14183224

A child is selected at random. Given that they chose rappelling, find the probability that the child is over 14.

  • A24.7%
  • B75%
  • C48%
  • D51%
  • E19.8%

Q2:

Ethan and Elizabeth are running for the presidency of the Students’ Union at their school. The votes they received from each of 3 classes are shown in the table. What is the probability that a student voted for Elizabeth given that they are in the Class B?

Class AClass BClass CTotal
Ethan161169177507
Elizabeth147195152494
  • A3877
  • B1528
  • C411
  • D1577

Q3:

The two-way table shows the ages and activity choices of a group of children at a summer camp.

SwimmingClimbingRappeling
14 and Under15248
Over 14183224

A child is selected at random. Given that this child is over 14, find the probability, to the nearest percent, that they chose climbing.

  • A43%
  • B36%
  • C57%
  • D76%
  • E26%

Q4:

The table below contains data from a survey of core gamers who were asked whether their preferred gaming platform was the smartphone, the console, or the PC. The gamers are split by gender.

Find the probability that a core gamer chosen at random prefers using a console. Give your answer to three decimal places.

Given that a core gamer prefers to play using a console, find the probability that they are male. Give your answer to three decimal places.

Q5:

A fanzine website for the TV show A Maze in Space collects data on the number of new alien species encountered by two starships in each season of the show. The data for seasons 1, 2, and 7 are shown in the table below, split between the two starships: Zeta and Geoda.

A Maze in Space
New Species Encountered
Season 1Season 2Season 7Total
Starship Zeta33511
Starship Geoda2836872
Total31391383

Find the probability that a new alien species chosen at random was encountered by starship Geoda. Give your answer to three decimal places.

Q6:

A fanzine website for the TV show A Maze in Space collects data on the number of new alien species encountered by two starships in each season of the show. The data for seasons 1, 2, and 7 are shown in the table below, split between two starships: Zeta and Geoda.

A Maze in Space
New Species Encountered
Season 1Season 2Season 7Total
Starship Zeta33511
Starship Geoda2836872
Total31391383

Given that a new alien species was encountered in season 7, find the probability that they were encountered by Starship Geoda. Give your answer to three decimal places.

Q7:

Data is collected from the TV show AMaze in Space on the number of new alien species first contact is made with. The data for Starship Zeta in seasons 1, 2, and 7 are shown in the table below. The data have also been categorized by whether the crew member who made first contact was male or female.

From the table, find the probability that first contact was made with a new alien species by a female crew member. Give your answer to three decimal places.

Find the probability that first contact was made in season 1 and by a female crew member. Give your answer to three decimal places.

Given that first contact was made with an alien species chosen at random, in season 1, find the probability that first contact was made by a female crew member. Give your answer to three decimal places.

Are the events “S1 = first contact made in season 1” and “female” independent?

  • AYes, they are independent.
  • BNo, they are not independent.

Q8:

Benjamin spins two spinners. The first has six equal sectors numbered from 1 to 6, and the second has four equal sectors numbered 1 to 4. He draws a two-way table to represent the sample space, as shown in the figure.

Work out the probability that at least one of the spinners lands on a 2.

  • A512
  • B14
  • C15
  • D16
  • E38

Work out the probability that the sum of the numbers is even.

  • A12
  • B23
  • C14
  • D1124
  • E524

Work out the probability that at least one of the spinners lands on a 2 and the sum of the numbers is even.

  • A14
  • B78
  • C316
  • D18
  • E16

Work out the probability that the sum of the numbers is even given that at least one of the spinners lands on a 2.

  • A16
  • B49
  • C25
  • D13
  • E18

Q9:

In an experiment, James is going to spin a fair three-sided spinner and a fair four-sided spinner. He draws a two-way table to show all of the possible outcomes.

1234
1(1, 1)(1, 2)(1, 3)(1, 4)
2(2, 1)(2, 2)(2, 3)(2, 4)
3(3, 1)(3, 2)(3, 3)(3, 4)

In his experiment, he wants to look at two events: spinning two numbers whose sum is prime, 𝐴, and spinning at least one three, 𝐵.

Find 𝑃(𝐴).

  • A34
  • B12
  • C23
  • D512
  • E712

Find 𝑃(𝐵).

  • A512
  • B13
  • C12
  • D23
  • E712

Find 𝑃(𝐴𝐵).

  • A23
  • B12
  • C13
  • D512
  • E14

Find 𝑃(𝐵𝐴).

  • A47
  • B27
  • C13
  • D14
  • E37

Find 𝑃(𝐴𝐵).

  • A14
  • B16
  • C12
  • D13
  • E5114

Is it true that 𝑃(𝐴)𝑃(𝐵𝐴)=𝑃(𝐴𝐵) and 𝑃(𝐵)𝑃(𝐴𝐵)=𝑃(𝐴𝐵)?

  • AYes
  • BNo

Q10:

Two biased coins were tossed. Coin 1 was tossed 100 times, 18% of which were heads. Coin 2 was tossed 300 times, 30% of which were heads.

If we chose a trial at random and found it to be a tail, find the probability of each of the following events.

The selected coin was coin 1. Round your answer to three decimal places.

The selected coin was coin 2. Round your answer to three decimal places.

Q11:

You have two boxes. Box 1 contains 5 red, 2 black, and 3 white balls. Box 2 contains 3 red, 10 black, and 14 white balls. If we select a ball at random, find the probability of each of the following events.

The selected ball is drawn from box 1, given that it is a black ball. Round your answer to two decimal places.

The selected ball is not a black ball if it is drawn from box 2. Round your answer to two decimal places.

Q12:

An employee received a total of 800 emails in two different email accounts. 25% of these messages were in account 1. 2% of the messages received in account 1 and 13% of the messages received in account 2 were spam.

If a selected message is not spam, find the probability that it is from account 2. Round your answer to three decimal places.

Q13:

For a group of 500 patients, 50 of them have normal blood pressure and the rest have high blood pressure. 100 of the patients are overweight. 20% of those who have high blood pressure are overweight.

Find the values of 𝑎, 𝑏, 𝑐, and 𝑑 in the following table.

OverweightNot Overweight
High Blood Pressure𝑎𝑏
Normal Blood Pressure𝑐𝑑
  • A𝑎=100, 𝑏=175, 𝑐=50, 𝑑=175
  • B𝑎=175, 𝑏=175, 𝑐=100, 𝑑=50
  • C𝑎=175, 𝑏=175, 𝑐=50, 𝑑=100
  • D𝑎=90, 𝑏=360, 𝑐=10, 𝑑=40
  • E𝑎=90, 𝑏=360, 𝑐=100, 𝑑=175

An overweight patient is selected at random. Find the probability that this patient has normal blood pressure.

Q14:

Consider the two-way table showing how many men and women have pets and how many do not.

Has a PetDoes Not Have a Pet
Men2266
Women7834

Find the probability that someone has pets and that they are a woman.

  • A1750
  • B17100
  • C3950
  • D39100
  • E1150

Q15:

Two boxes contain a number of defective, partially defective (failing after a couple of hours of use), and acceptable light bulbs.

The numbers are given in the table.

Box 1Box 2
Defective123
Partially Defective322
Acceptable2540

A light bulb is chosen at random and put to use. If it does not fail immediately, what is the probability that it is chosen from box 2? Round your answer to three decimal places.

Q16:

The following table shows the number of games played by three teams.

Team 1Team 2Team 3
Won141510
Lost632

Find the probability that one of the games won was played by team 2. Round your answer to two decimal places.

Q17:

Each of 1,000 voters elected one of two candidates, 𝐴 or 𝐵. The number of valid votes is 980, and 392 of them were for 𝐵.

If we selected a voter at random, find the probability that their vote was invalid.

If we selected a vote and it was found to be valid, find the probability that the voter elected candidate 𝐴.

Q18:

At a school, each student in the arts studies section was required to select one of two complementary subjects, chemistry or mathematics, and pass its exam.

In a group of 400 students, 100 students chose mathematics. The number of students who passed the mathematics exam is 55, and the number of students who passed the chemistry exam is 160.

A student was chosen at random and was found to have failed to meet this requirement.

Find the probability that the student had selected mathematics.

Round your answer to the nearest two decimal places.

Find the probability that the student had selected chemistry.

Round your answer to the nearest two decimal places.

Q19:

Students’ grades on a test were ranked into 3 levels, A, B, and C. Two different groups of students were selected, each containing 50 students. In group 1, 20% of the students got an A and 38% got a B. In group 2, 20% of the students got an A and 30% got a B.

If a student is selected at random, find the probability of each of the following events.

The selected student got a C.

The selected student got a C, given that they were selected from group 1.

Q20:

The following table shows the numbers of ice creams sold in an ice cream shop.

ConeCup
Chocolate Flavor12020
Vanilla Flavor9020

If we select an ice cream at random and find it to be sold in a cup, find the probability that it is chocolate flavored.

For a randomly selected ice cream, find the probability that it is sold in a cone or that it is vanilla flavored.

Q21:

Consider the two-way table showing how many men and women have laptops and how many do not.

Has a LaptopDoes Not Have a Laptop
Men3066
Women7034

Find the probability that someone has a laptop and that they are a man.

Q22:

The following table shows the number of airplanes that need repairs and the number of airplanes that do not need repairs at a certain airline.

Need RepairsDo Not Need Repairs
2-Engine Planes1228
4-Engine Planes516

If you selected a 4-engine plane, find the probability that it does not need repairs. Round your answer to 3 decimal places.

Q23:

A company manufactures a product in two different plants, 𝑃 and 𝑃. The company supplies three customers, 𝐶, 𝐶, and 𝐶 equally, each with 80 units a month. 𝑃 produces 10 units of this product per month and the company distributes this amount among the three customers 𝐶, 𝐶, and 𝐶 in percentages of 20%, 30%, and 50% respectively. If you select a unit at random from a 𝐶 outlet, find the probability that it is produced by 𝑃.

Q24:

A teacher conducted a survey in his class about what his students prefer to do on the weekend. The following table describes the data of the survey.

GirlsBoys
Watching TV74
Playing Football212
Going to the Movie Theater83

If a student is chosen at random, what is the probability that the chosen student prefers watching TV if you know that she is a girl?

  • A13
  • B736
  • C717
  • D12
  • E1136

Q25:

The table below shows data from a survey in which customers were asked about their level of satisfaction with the customer service. The customers are split by gender.

SatisfiedNot SatisfiedNeutralTotal
Men2719955
Women20121345

If a customer is satisfied, find the probability that they are a woman. Round your answer to two decimal places.

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