Worksheet: Addition Rule for Probability

In this worksheet, we will practice finding the probability of unions and intersections of events using the addition rule for probability.

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 𝐴 and 𝐡 are two events with probabilities 𝑃(𝐴)=57 and 𝑃(𝐡)=47. Given that 𝑃(𝐴βˆͺ𝐡)=67, determine 𝑃(π΄βˆ’π΅).

  • A715
  • B37
  • C57
  • D27
  • E79

Q5:

Suppose 𝑋 and π‘Œ are two events with probabilities 𝑃(𝑋)=0.49 and 𝑃(π‘Œ)=0.48. Given that 𝑃(𝑋βˆͺπ‘Œ)=0.95, determine 𝑃(π‘‹βˆ©π‘Œ).

Q6:

Suppose 𝐴 and 𝐡 are two events. Given that π΅βŠ‚π΄, 𝑃(𝐡)=49, and 𝑃(π΄βˆ’π΅)=15, determine 𝑃(𝐴).

  • A245
  • B2845
  • C2945
  • D845
  • E619

Q7:

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

Q8:

Suppose 𝐴 and 𝐡 are two events with probabilities 𝑃(𝐴)=0.6 and 𝑃(𝐡)=0.5. Given that 𝑃(𝐴βˆͺ𝐡)=0.3, determine the probability that only one of the events 𝐴 and 𝐡 occurs.

Q9:

A ball is drawn at random from a bag containing 12 balls each with a unique number from 1 to 12. Suppose 𝐴 is the event of drawing an odd number and 𝐡 is the event of drawing a prime number. Find 𝑃(π΄βˆ’π΅).

  • A112
  • B512
  • C13
  • D16
  • E12

Q10:

Suppose 𝐴 and 𝐡 are events in the sample space of an experiment. Given that 𝑃(𝐴′)=12 and 𝑃(π΄β€²βˆ©π΅β€²)=112, find the value of 𝑃(π΄β€²βˆ’π΅β€²).

  • A13
  • B12
  • C512
  • D16
  • E34

Q11:

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

  • A𝐡
  • Bβˆ…
  • C𝐴
  • Dπ΄βˆ’π΅

Q12:

𝐴 and 𝐡 are two events in a sample space of a random experiment where 𝑃(𝐴)=310, 𝑃(𝐡)=15, and 𝑃(π΄βˆ’π΅)=110. Find 𝑃(𝐴βˆͺ𝐡).

  • A35
  • B15
  • C310
  • D110

Q13:

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

  • A𝐡
  • B𝐴∩𝐡
  • C𝐴
  • Dβˆ…

Q14:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃(𝐴)=58, 𝑃(𝐡)=34, and 𝑃(π΄βˆ’π΅)=14, find 𝑃𝐴βˆͺ𝐡.

  • A34
  • B54
  • C58
  • D18
  • E12

Q15:

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

Q16:

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

Q17:

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

Q18:

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

Q19:

Suppose 𝐴 and 𝐡 are two events in a random experiment. Given that 𝑃(𝐡)=710𝑃(𝐴), 𝑃(π΄βˆ’π΅)=0.12, and π‘ƒο€Ίπ΅βˆ©π΄ο†=0.03, find 𝑃(𝐡).

Q20:

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

Q21:

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

Q22:

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

Q23:

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

Q24:

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).

Q25:

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).

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