Worksheet: Tree Diagrams

In this worksheet, we will practice drawing and using tree diagrams.

Q1:

Write the sample space 𝑆 for whether a regular coin lands heads side up (H) or tails side up (T) when flipped.

  • A 𝑆 = { , } H T
  • B 𝑆 = { , } H T T H
  • C 𝑆 = { , } H H
  • D 𝑆 = { , } T T H H
  • E 𝑆 = { , } T T

Q2:

A coin is flipped three consecutive times. Which of the following represents the event of getting a head on the first flip?

  • A 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H H T T
  • B 𝐴 = { ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H
  • C 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } T H H T H T T T H T T T
  • D 𝐴 = { ( , , ) } H H H

Q3:

A coin is flipped 3 times. Use the following tree diagram to determine the event 𝐴 that includes all of the outcomes where no more than two tails are flipped.

  • A 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H H T T T H H T H T T T H
  • B 𝐴 = { ( , , ) , ( , , ) , ( , , ) } H T T T H T T T H
  • C 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } H T T T H T T T H T T T
  • D 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H T H H

Q4:

If two spinners are spun, where the first is numbered from 1 to 2 and the second from 1 to 9, determine the probability of both spinners stopping at even numbers using a tree diagram.

  • A9 out of 11
  • B4 out of 18
  • C8 out of 18
  • D5 out of 11
  • E14 out of 18

Q5:

Madison has drawn a tree diagram to represent the sample space of tossing two coins.

By extending the tree diagram or otherwise, find the number of outcomes in the sample space of the experiment of tossing two coins and spinning two spinners: one with 10 equal sections labeled 1 to 10 and one with 12 equal sections labeled 1 to 12.

Q6:

A pizza parlor offers customizable pizzas. Customers can choose one of two options for the base, cheese, and sauce and then add an optional topping.

Base Cheese Sauce Topping (at most one)
Thin crust or deep pan Cheddar or mozzarella Tomato or barbecue Pepperoni, pineapple, or mushroom

Using tree diagrams, or otherwise, represent all the possible pizza combinations and determine how many different pizzas there are.

Q7:

Suppose two spinners are spun. The first spinner is numbered from 1 to 3 and the second from 1 to 9. Using a tree diagram, determine what the probability of the sum of both spins being greater than 4 is.

  • A24 out of 27
  • B6 out of 27
  • C21 out of 27
  • D3 out of 36

Q8:

A fair coin is flipped three times and the results are recorded, writing H for heads and T for tails. Which of the following represents the event of getting no heads?

  • A 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H T H T H H T T T H H T H T T T H T T T
  • B 𝐴 = { ( , , ) } T T T
  • C 𝐴 = { ( , , ) } H H H
  • D 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H H T T T H H T H T T T H

Q9:

Which of the following describes the sample space of a random experiment?

  • AAn experiment which we can determine all its possible outcomes.
  • BThe set of all impossible outcomes of a random experiment.
  • CThe set of some possible outcomes of a random experiment.
  • DThe set of all possible outcomes of a random experiment.

Q10:

I write a two-digit number by randomly picking each digit to be either 3 or 7, where digits can be repeated. Write the set of all possible outcomes.

  • A { 3 7 , 3 3 , 7 3 , 7 7 }
  • B { 3 7 3 , 7 3 7 }
  • C { 3 7 , 7 3 }
  • D { 3 3 , 7 7 }
  • E { 3 7 3 , 7 3 7 , 3 3 7 , 7 7 3 }

Q11:

A bag contains 7 balls which are numbered 1 to 7. Determine the sample space of choosing a ball at random.

  • A { 1 , 2 , 6 , 4 , 5 , 7 }
  • B { 1 , 2 , 3 , 4 , 5 , 6 , 7 }
  • C { 3 , 1 , 2 , 5 , 4 , 7 }
  • D { 1 , 5 , 6 , 2 , 3 , 7 , 8 }

Q12:

A coin is flipped three times. Which of the following is the event that at least two tails appear?

  • A 𝐴 = { ( , , ) , ( , , ) , ( , , ) } H T T T H T T T H
  • B 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } H T T T H T T T H T T T
  • C 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H T H H
  • D 𝐴 = { ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) , ( , , ) } H H H H H T H T H H T T T H H T H T T T H

Q13:

The given probability tree shows two events: 𝐴 and 𝐵. 𝐴 is the event of it raining and 𝐵 is the event of a group of friends playing football.

Determine the probability of the friends playing football given that it rains.

Determine the probability of the friends playing football and it raining.

Determine the probability of the group of friends playing football irrespective of whether it rains.

Q14:

A bag contains 6 balls numbered from 60 to 65. Which of the following is the sample space for the experiment of choosing a numbered ball at random?

  • A 𝑆 = { 6 0 , 6 3 }
  • B 𝑆 = { 6 0 , 6 5 }
  • C 𝑆 = { 6 0 , 6 1 , 6 2 , 6 3 , 6 4 , 6 5 }
  • D 𝑆 = { 6 1 , 6 2 , 6 3 , 6 4 , 6 5 , 6 6 }
  • E 𝑆 = { 5 9 , 6 0 , 6 1 , 6 2 , 6 3 , 6 4 }

Q15:

A fair coin is flipped twice and the results are recorded, writing H for heads and T for tails. Which of the following represents the sample space of this random experiment?

  • A { , , , } H H H T H T T T
  • B { , } H H T T
  • C { , , , } H H H T T H T T
  • D { , } H T T H

Q16:

Which of the following represents the sample space of choosing a digit at random from the number 148,250?

  • A 𝑆 = { 1 , 2 , 4 , 5 }
  • B 𝑆 = { 2 , 4 , 5 }
  • C 𝑆 = { 0 , 1 , 2 , 4 , 5 , 8 }
  • D 𝑆 = { 0 , 1 , 2 , 4 , 5 }

Q17:

Which experiment can be modeled by this tree diagram?

  • ATossing a coin eight times
  • BTossing eight coins
  • CTossing a coin three times
  • DTossing three coins

Q18:

Upon rolling a die and flipping a coin once, 𝐴 is the event of getting tails and an even number, 𝐵 is the event of getting heads and an odd number, and 𝐶 is the event of the nonoccurence of 𝐴 or the nonoccurence of 𝐵. Determine the outcomes in the event 𝐶.

  • A 𝐶 = { ( , 2 ) , ( , 4 ) , ( , 6 ) , ( , 1 ) , ( , 3 ) , ( , 5 ) } T T T H H H
  • B 𝐶 = { ( , 2 ) , ( , 4 ) , ( , 6 ) , ( , 1 ) , ( , 3 ) , ( , 5 ) } H H H T T T
  • C 𝐶 = { ( , 1 ) , ( , 2 ) , ( , 3 ) , ( , 4 ) , ( , 5 ) , ( , 6 ) , ( , 1 ) , ( , 2 ) , ( , 3 ) , ( , 4 ) , ( , 5 ) , ( , 6 ) } H H H H H H T T T T T T
  • D 𝐶 =

Q19:

A fair coin is flipped and a fair die is rolled. Which of the following represents the event of getting a head on the coin and rolling and odd number on the die?

  • A 𝐴 = { ( , 2 ) , ( , 4 ) , ( , 6 ) } H H H
  • B 𝐴 = { ( , 1 ) , ( , 3 ) , ( , 5 ) } T T T
  • C 𝐴 = { ( , 1 ) , ( , 1 ) , ( , 3 ) , ( , 3 ) , ( , 5 ) , ( , 5 ) } T H T H T H
  • D 𝐴 = { ( , 1 ) , ( , 3 ) , ( , 5 ) } H H H

Q20:

List all the possible outcomes when spinning this spinner.

  • A 𝑃 , 𝑇
  • B 𝑃 , 𝑇 , 𝑉
  • C 𝑃
  • D 𝑃 , 𝑇 , 𝑉 , 𝑈 , 𝑄
  • E 𝑃 , 𝑇 , 𝑉 , 𝑈

Q21:

Daniel is going to spin both of these spinners and record the two letters that appear.

Which of the following tree diagrams shows the sample space of the letter pairs?

  • A
  • B
  • C
  • D

How many possible outcomes are there?

Find the probability of getting 𝐴 or 𝐵 on spinner 1 and 𝐷 on spinner 2. Give your answer as a fraction in its simplest form.

  • A 1 1 2
  • B 1 3
  • C 1 6
  • D 1 9

Q22:

A class have a die with the numbers 1, 2, 3, 4, 5, and 6 and a spinner with the numbers 2, 4, 6, and 8. They are going to roll the die and spin the spinner and find the sum of the two numbers that appear. They started to complete a two-way table to model the sample space.

Die
1 2 3 4 5 6
Spinner 2 3 4 5 6 7 8
4 5 6 7 8
6 7 8
8 9 10

How many possible outcomes are there?

By completing the table, find the probability of the sum being a number greater than 10.

Give your answer as a fraction in its simplest form.

  • A 2 5
  • B 1 3
  • C 1 6
  • D 1 4
  • E 1 5

Q23:

In an experiment, this spinner is spun. List all the possible outcomes.

  • A 2 , 6
  • B 2 , 6 , 5
  • C 2 , 6 , 5 , 8
  • D 2 , 6 , 5 , 8 , 3 , 4
  • E 2 , 6 , 5 , 8 , 3

Q24:

If the shown spinner was spun once and a coin was tossed once, determine the number of possible outcomes.

Q25:

The image shows a fair spinner.

Daniel wanted 5 random numbers from 1 to 5, so he spun the spinner 5 times. His results were 1, 2, 3, 4, and 5.

Which of the following statements is true about his results?

  • AThey are random numbers because the spinner is fair and was spun randomly.
  • BThey are random numbers because you would expect each number to appear once.
  • CThey are not random numbers because you would expect some numbers to come up more than others.
  • DThey are random numbers because the spinner is fair and he got one of each number.
  • EThey are not random numbers because the results came in order of size, which is unlikely.

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