Lesson Worksheet: Operations on Events Mathematics

In this worksheet, we will practice finding the probabilities of the complement, intersection, and union of events.

Q1:

Denote by 𝐴 and 𝐡 two events with probabilities 𝑃(𝐴)=0.2 and 𝑃(𝐡)=0.47. Given that 𝑃(𝐴∩𝐡)=0.18, find 𝑃(𝐴βˆͺ𝐡).

Q2:

Denote by 𝐴 and 𝐡 two events with probabilities 𝑃(𝐴)=0.58 and 𝑃(𝐡)=0.2. Given that 𝑃(𝐴βˆͺ𝐡)=0.64, find 𝑃(𝐴∩𝐡).

Q3:

Suppose 𝐴 and 𝐡 are two events with probability 𝑃(𝐴)=0.6 and 𝑃(𝐡)=0.5. Given that 𝑃(𝐴∩𝐡)=0.4, what is the probability that at least one of the events does not occur?

Q4:

Suppose π‘₯ and 𝑦 are two events with probabilities 𝑃(π‘₯)=0.49 and 𝑃(𝑦)=0.48. Given that 𝑃(π‘₯βˆͺ𝑦)=0.95, determine 𝑃(π‘₯βˆ©π‘¦).

Q5:

Suppose that 𝑋 and π‘Œ are two events with probabilities 𝑃(π‘Œ)=13 and 𝑃(𝑋)=𝑃𝑋. Given that 𝑃(π‘‹βˆ©π‘Œ)=18, determine 𝑃(𝑋βˆͺπ‘Œ).

  • A724
  • B524
  • C1332
  • D1724
  • E56

Q6:

Suppose 𝐴 and 𝐡 are events such that π΅βŠ‚π΄. Determine 𝐴βˆͺ𝐡.

  • A𝐡
  • B𝐴∩𝐡
  • C𝐴
  • Dβˆ…

Q7:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃(𝐡)=58,𝑃(𝐴βˆͺ𝐡)=34, and π΅βŠ‚π΄, find 𝑃(𝐴).

  • A34
  • B18
  • C38
  • D58

Q8:

A group of 68 school children completed a survey asking about their television preferences. The results show that 43 of the children watch channel 𝐴, 26 watch channel 𝐡, and 12 watch both channels. If a child is selected at random from the group, what is the probability that they watch at least one of the two channels?

  • A4368
  • B734
  • C1334
  • D3168
  • E5768

Q9:

Suppose 𝐴 and 𝐡 are events. Given that π΄βŠ‚π΅, 𝑃(𝐴)=π‘₯, 𝑃𝐡=7π‘₯, and 𝑃(𝐴βˆͺ𝐡)=7π‘₯+0.4, find the value of π‘₯.

  • A710
  • B37
  • C17
  • D910
  • E110

Q10:

Suppose 𝐴 and 𝐡 are events. Given that 𝑃(𝐴)=4π‘₯, 𝑃𝐡=π‘₯,𝑃(𝐴βˆͺ𝐡)=3π‘₯+0.9, and 𝑃(𝐴∩𝐡)=12π‘₯, find the value of π‘₯.

  • A35
  • B14
  • C15
  • D519

Q11:

Suppose 𝐴 and 𝐡 are two events with probabilities 𝑃(𝐴)=25 and 𝑃(𝐡)=π‘₯. Given that 𝑃(𝐴βˆͺ𝐡)=13 and π΄βŠ‚π΅, find the value of π‘₯.

  • A115
  • B25
  • C13
  • D23
  • E415

Q12:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃(𝐴βˆͺ𝐡)=0.64 and π΄βŠ‚π΅, find 𝑃(𝐡).

Q13:

A bag contains 15 blue balls and 20 red balls. A ball is chosen at random and the color is recorded. The ball is then replaced and another ball is chosen at random from the bag. What is the probability of both chosen balls being blue?

  • A1649
  • B38119
  • C47
  • D949

Q14:

Suppose that 𝐴 and 𝐡 are two mutually exclusive events. Given that 𝑃(𝐡)=0.01 and 𝑃(𝐴βˆͺ𝐡)=0.62, determine 𝑃(𝐴).

Q15:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃𝐡=0.35, 𝑃(𝐴βˆͺ𝐡)=0.86, and 𝑃(𝐴∩𝐡)=𝑃(𝐴)×𝑃(𝐡), find 𝑃(𝐴).

Q16:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃(𝐴)=0.37, 𝑃(𝐴βˆͺ𝐡)=0.73, and 𝑃(𝐴∩𝐡)=0.19, determine 𝑃(𝐡).

Q17:

Suppose that 𝐴 and 𝐡 are events with probabilities 𝑃(𝐴)=0.64 and 𝑃(𝐡)=π‘š. Given that 𝑃(𝐴βˆͺ𝐡)=0.8 and 𝑃(𝐴∩𝐡)=0.55, find the value of π‘š.

Q18:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃(𝐡)=3𝑃(𝐴), 𝑃(𝐴βˆͺ𝐡)=0.93, and π΄βŠ‚π΅, determine 𝑃(𝐴).

Q19:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃(𝐡)=0.06, 𝑃(𝐴βˆͺ𝐡)=0.09, and π΅βŠ‚π΄, determine 𝑃(𝐴).

Q20:

Suppose 𝐴 and 𝐡 are two events with probabilities 𝑃(𝐴)=58 and 𝑃(𝐡)=12. Given that 𝑃(π΄βˆ’π΅)=14, find π‘ƒο€Ίπ΅βˆ©π΄ο†.

  • A14
  • B18
  • C38
  • D34

Q21:

The diagram represents the sample space 𝑆 and the events 𝐴, 𝐡, and 𝐢. Determine 𝑃(𝐴∩𝐡).

  • A23
  • B16
  • C12
  • D1

Q22:

Out of a group of 100 people, 46 have dogs, 41 have cats, and 28 have rabbits. 12 of the people have both dogs and cats, 10 have both cats and rabbits, and 9 have both dogs and rabbits. 8 of the people have dogs, cats, and rabbits.

Find the probability of randomly selecting a person who has dogs, cats, and rabbits. Give your answer as a fraction in its simplest form.

  • A13100
  • B110
  • C225
  • D9100
  • E325

Find the probability of randomly selecting a person who only has dogs and rabbits. Give your answer as a fraction in its simplest form.

  • A11100
  • B3100
  • C1100
  • D120
  • E9100

Find the probability of selecting a person who has pets. Give your answer as a fraction in its simplest form.

  • A1320
  • B59100
  • C2325
  • D1
  • E2320

Find the probability of selecting a person who does not have pets. Give your answer as a fraction in its simplest form.

  • A2325
  • B110
  • C320
  • D225
  • E0

Q23:

If 𝑁 and 𝑂 are events and 𝑁 is a subset of 𝑂 then which of the following is equal to 𝑃(π‘βˆ©π‘‚)?

  • A𝑃(𝑂)
  • B0
  • C1
  • D𝑃(𝑁)

Q24:

Given that 𝐴 and 𝐡 are two events in the sample space of a random experiment, where π΄βŠ‚π΅, determine 𝐴∩𝐡.

  • A𝐡
  • B𝐴βˆͺ𝐡
  • C𝐴
  • Dβˆ…

Q25:

I write a two-digit number by randomly picking each digit from the set {3,5,6,7,8,9}, where digits can be repeated. Write the set that represents the event that the number is divisible by 3 and 6.

  • A{75}
  • B{36,66,78,96}
  • C{36,56,68,76,88,96}
  • D{56,88,96}
  • E{33,36,39,57,63,66,69,75,78,87,93,96,99}

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