Lesson Worksheet: Counting Outcomes Mathematics

In this worksheet, we will practice determining and counting the possible outcomes using tables and tree diagrams to find the probability of a specific event.


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.

BaseCheeseSauceTopping (at Most One)
Thin crust or deep panCheddar or mozzarellaTomato or barbecuePepperoni, pineapple, or mushroom

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


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

  • A𝐴={(,,),(,,),(,,),(,,)}HHHHHTHTHHTT
  • B𝐴={(,,),(,,),(,,)}HHHHHTHTH
  • C𝐴={(,,),(,,),(,,),(,,)}THHTHTTTHTTT
  • D𝐴={(,,)}HHH


Khalid 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.

  • A112
  • B13
  • C16
  • D19


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.

  • A𝐴={(,,),(,,),(,,),(,,),(,,),(,,),(,,)}HHHHHTHTHHTTTHHTHTTTH
  • B𝐴={(,,),(,,),(,,)}HTTTHTTTH
  • C𝐴={(,,),(,,),(,,),(,,)}HTTTHTTTHTTT
  • D𝐴={(,,),(,,),(,,),(,,)}HHHHHTHTHTHH


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{37,33,73,77}
  • B{373,737}
  • C{37,73}
  • D{33,77}
  • E{373,737,337,773}


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

  • A𝑆={,}HT
  • B𝑆={,}HTTH
  • C𝑆={,}HH
  • D𝑆={,}TTHH
  • E𝑆={,}TT


Abeer 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.


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.

  • A𝐶={(,2),(,4),(,6),(,1),(,3),(,5)}TTTHHH
  • B𝐶={(,2),(,4),(,6),(,1),(,3),(,5)}HHHTTT
  • C𝐶={(,1),(,2),(,3),(,4),(,5),(,6),(,1),(,2),(,3),(,4),(,5),(,6)}HHHHHHTTTTTT
  • D𝐶=


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?

  • A𝐴={(,,),(,,),(,,),(,,),(,,),(,,),(,,)}HHTHTHHTTTHHTHTTTHTTT
  • B𝐴={(,,)}TTT
  • C𝐴={(,,)}HHH
  • D𝐴={(,,),(,,),(,,),(,,),(,,),(,,),(,,)}HHHHHTHTHHTTTHHTHTTTH


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}

Practice Means Progress

Download the Nagwa Practice app to access 15 additional questions for this lesson!

scan me!

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.