Worksheet: Sample Space and Probability

In this worksheet, we will practice determining the sample space and the probability.

Q1:

A card is drawn at random from a pack of cards numbered from 1 to 32. Write the set of outcomes that the number on the card is odd and divisible by 3.

  • A { 1 , 3 , 5 , 7 , 9 , 1 1 , 1 3 , 1 5 , 1 7 , 1 9 , 2 1 , 2 3 , 2 5 , 2 7 , 2 9 , 3 1 }
  • B { 6 , 1 2 , 1 8 , 2 4 , 3 0 }
  • C { 2 , 4 , 6 , 8 , 1 0 , 1 2 , 1 4 , 1 6 , 1 8 , 2 0 , 2 2 , 2 4 , 2 6 , 2 8 , 3 0 , 3 2 }
  • D { 3 , 9 , 1 5 , 2 1 , 2 7 }
  • E { 3 , 6 , 9 , 1 2 , 1 5 , 1 8 , 2 1 , 2 4 , 2 7 , 3 0 }

Q2:

A bag contains 4 red balls and 6 brown balls. Which of the following represents the sample space of choosing a ball at random and recording its colour?

  • A{brown}
  • B{red}
  • C{red, red}
  • D{red, brown}
  • E{brown, brown}

Q3:

Which of the following represents the sample space of flipping a coin and rolling a die?

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

Q4:

Which of the following is the definition of a sample space?

  • A the set of some possible outcomes of a random experiment
  • B one of the possible outcomes of a random experiment
  • C the set of all the possible outcomes of a random experiment

Q5:

Which of the following represents the sample space of choosing at random a prime number between 59 and 93?

  • A 𝑆 = { 7 1 , 7 3 , 7 9 , 8 3 , 8 9 }
  • B 𝑆 = { 5 9 , 6 1 , 6 7 , 7 1 , 7 3 , 7 9 , 8 3 , 8 9 }
  • C 𝑆 = { 6 7 , 7 1 , 7 3 , 7 9 , 8 3 }
  • D 𝑆 = { 6 1 , 6 7 , 7 1 , 7 3 , 7 9 , 8 3 , 8 9 }
  • E 𝑆 = { 7 1 , 7 3 , 7 9 , 8 3 }

Q6:

In a random experiment, a ball is chosen at random from a box that contains four balls: one blue, one purple, one brown, and one white. List all the possible outcomes of this experiment.

  • A{blue, white, grey, purple}
  • B{blue, purple, brown}
  • C{brown, white, purple}
  • D{purple, blue, brown, white}
  • E{purple, white, brown, orange}

Q7:

Which of the following denotes the sample space of randomly selecting an even number between 39 and 52?

  • A 𝑆 = { 3 8 , 4 0 , 4 2 , 4 4 , 4 6 , 4 8 , 5 0 , 5 2 }
  • B 𝑆 = { 4 0 , 4 2 , 4 4 , 4 6 , 4 8 , 5 0 , 5 2 }
  • C 𝑆 = { 4 0 , 4 4 , 4 8 }
  • D 𝑆 = { 4 0 , 4 2 , 4 4 , 4 6 , 4 8 , 5 0 }
  • E 𝑆 = { 4 2 , 4 4 , 4 6 , 4 8 , 5 0 }

Q8:

A fair coin is flipped and and a fair dice is rolled. Which of the following represents the event of rolling a number which is divisible by 3?

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

Q9:

A fair coin is flipped and a fair die is rolled. Which of the following represents the event of the coin landing tail-side up and an even number on the die?

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

Q10:

Two fair dice are rolled. Which of the following represents the event of rolling the same number on both dice?

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

Q11:

Two distinct digits are chosen at random from the set { 1 , 2 , 4 , 8 } . What is the probability that either of the digits is an odd number?

  • A 3 8
  • B 1 2
  • C 9 1 6
  • D 7 1 6
  • E 5 1 6

Q12:

I write a two-digit number by randomly picking each digit from the set { 2 , 3 , 4 , 5 , 7 , 8 } , where a digit can be repeated. Write the set of outcomes that is the event that the number is prime.

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

Q13:

In an experiment a coin is flipped repeatedly until the first time it lands heads side up, or until it lands tails side up three times in a row. Determine the set for event 𝐴 , which is the appearance of at least two tails.

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

Q14:

In an experiment a coin is repeatedly flipped until it lands head side up, or it lands tail side up three times in a row. Determine the set representing event 𝐴 , which is the appearance of at least one head.

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

Q15:

Given that the sample space is the set of all 2-digit numbers whose digits are different and belong to { 4 , 6 , 8 , 9 } , write the event that the number is odd.

  • A { 9 4 , 9 6 , 9 8 , 9 9 }
  • B { 9 4 , 9 6 , 9 8 }
  • C { 4 4 , 4 6 , 4 8 , 6 4 , 6 6 , 6 8 , 8 4 , 8 6 , 8 8 , 9 4 , 9 6 , 9 8 }
  • D { 4 9 , 6 9 , 8 9 }
  • E { 4 9 , 6 9 , 8 9 , 9 9 }

Q16:

What is the probability of choosing at random, two numbers which sum to 9, from the set { 1 , 5 , 7 , 8 } ? You are allowed to pick the same number more than once.

  • A 3 1 7
  • B 3 1 6
  • C 2 1 7
  • D 1 8
  • E 1 4

Q17:

I write a two-digit number by randomly picking each digit from the set { 2 , 4 , 8 } , where a digit can be repeated. What is the probability that the product of the two digits is 32?

  • A 3 1 0
  • B 1 3
  • C 1 5
  • D 2 9
  • E 4 9

Q18:

In an experiment a coin is repeatedly flipped until it lands head side up or it lands tail side up three times in a row. Determine the subset of the experiment’s sample space representing event 𝐴 , which is the appearance of at least two heads.

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

Q19:

A bag contains 20 identical cards numbered 1 to 20. A card is selected at random, and the number written on it is recorded. Write the set representing event 𝐴 , where the recorded number, π‘₯ , satisfies the inequality 5 π‘₯ βˆ’ 3 ≀ 1 7 .

  • A 𝐴 = { 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 }
  • B 𝐴 = { 1 , 2 , 3 }
  • C 𝐴 = { 4 }
  • D 𝐴 = { 1 , 2 , 3 , 4 }
  • E 𝐴 = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 }

Q20:

Two separate regular dice are rolled, and the numbers on their upper faces noted. Write down the sample space of the event 𝐴 , which is the appearance of two numbers such that one of them is twice the other.

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

Q21:

I write a two-digit number by randomly selecting two different digits from the set { 1 , 2 , 3 , 4 , 5 , 6 , 8 } . Which of the following represents the event that there is an odd digit in the tens position?

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

Q22:

I write a two-digit number by randomly picking each digit from the set { 3 , 4 , 9 } , where digits can be repeated. Which of the following is the event that the number is odd?

  • A { 3 4 , 4 4 , 9 4 }
  • B { 3 3 , 3 4 , 3 9 , 9 3 , 9 4 , 9 9 }
  • C { 3 4 , 9 4 }
  • D { 3 3 , 3 9 , 4 3 , 4 9 , 9 3 , 9 9 }
  • E { 3 9 , 4 3 , 4 9 , 9 3 }

Q23:

I write a two-digit number by randomly picking each digit from the set { 2 , 4 , 5 , 6 } , where digits can be repeated. Write the set that represents the event that the number is divisible by 7.

  • A { 2 4 , 4 4 , 5 2 , 5 6 , 6 4 }
  • B { 2 4 , 4 2 , 4 5 , 5 4 , 6 6 }
  • C { 2 5 , 4 5 , 5 5 , 6 5 }
  • D { 4 2 , 5 6 }
  • E { 2 4 , 4 2 , 5 4 , 6 6 }

Q24:

Write, as a subset of the sample space, the event of rolling a number greater than 1 on a fair die.

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

Q25:

A fair six-sided die has the numbers 1 to 6 on its faces. In an experiment, we roll the die and observe which number appears on the uppermost face. Write down the subset of the sample space for the event where the outcome is neither less than 2 nor greater than 4.

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

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