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 𝐵 two events with probabilities 𝑃(𝐴)=0.2 and 𝑃(𝐵)=0.47. Given that 𝑃(𝐴𝐵)=0.18, find 𝑃(𝐴𝐵).

Q2:

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

Q3:

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

Q4:

Suppose 𝐴, 𝐵, and 𝐶 are three mutually exclusive events in a sample space 𝑆. Given that 𝑆=𝐴𝐵𝐶, 𝑃(𝐴)=15𝑃(𝐵), and 𝑃(𝐶)=4𝑃(𝐴), find 𝑃(𝐵𝐶).

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

Q5:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃(𝐴)=57 and 𝑃(𝐵)=47. Given that 𝑃(𝐴𝐵)=67, determine 𝑃(𝐴𝐵).

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

Q6:

Suppose 𝑋 and 𝑌 are two events with probabilities 𝑃(𝑋)=0.49 and 𝑃(𝑌)=0.48. Given that 𝑃(𝑋𝑌)=0.95, determine 𝑃(𝑋𝑌).

Q7:

Suppose 𝐴 and 𝐵 are two events. Given that 𝐵𝐴, 𝑃(𝐵)=49, and 𝑃(𝐴𝐵)=15, determine 𝑃(𝐴).

  • A 2 4 5
  • B 2 8 4 5
  • C 2 9 4 5
  • D 8 4 5
  • E 6 1 9

Q8:

Suppose that 𝑋 and 𝑌 are two events with probabilities 𝑃(𝑌)=13 and 𝑃(𝑋)=𝑃𝑋. Given that 𝑃(𝑋𝑌)=18, determine 𝑃(𝑋𝑌).

  • A 7 2 4
  • B 5 2 4
  • C 1 3 3 2
  • D 1 7 2 4
  • E 5 6

Q9:

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

Q10:

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 𝐵 is the event of drawing a prime number. Find 𝑃(𝐴𝐵).

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

Q11:

Suppose 𝐴 and 𝐵 are events in the sample space of an experiment. Given that 𝑃(𝐴)=12 and 𝑃(𝐴𝐵)=112, find the value of 𝑃(𝐴𝐵).

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

Q12:

Given that 𝐴 and 𝐵 are two events in the sample space of a random experiment, where 𝐵𝐴, determine 𝐵𝐴.

  • A 𝐵
  • B
  • C 𝐴
  • D 𝐴 𝐵

Q13:

𝐴 and 𝐵 are two events in a sample space of a random experiment where 𝑃(𝐴)=310, 𝑃(𝐵)=15, and 𝑃(𝐴𝐵)=110. Find 𝑃(𝐴𝐵).

  • A 3 5
  • B 1 5
  • C 3 1 0
  • D 1 1 0

Q14:

Suppose 𝐴 and 𝐵 are events such that 𝐵𝐴. Determine 𝐴𝐵.

  • A 𝐵
  • B 𝐴 𝐵
  • C 𝐴
  • D

Q15:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃(𝐴)=58, 𝑃(𝐵)=34, and 𝑃(𝐴𝐵)=14, find 𝑃𝐴𝐵.

  • A 3 4
  • B 5 4
  • C 5 8
  • D 1 8
  • E 1 2

Q16:

Suppose that 𝐴 and 𝐵 are two events. Given that 𝑃(𝐵)=58,𝑃(𝐴𝐵)=34, and 𝐵𝐴, find 𝑃(𝐴).

  • A 3 4
  • B 1 8
  • C 3 8
  • D 5 8

Q17:

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 𝐵, 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?

  • A 4 3 6 8
  • B 7 3 4
  • C 1 3 3 4
  • D 3 1 6 8
  • E 5 7 6 8

Q18:

Suppose 𝐴 and 𝐵 are events. Given that 𝐴𝐵, 𝑃(𝐴)=𝑥, 𝑃𝐵=7𝑥, and 𝑃(𝐴𝐵)=7𝑥+0.4, find the value of 𝑥.

  • A 7 1 0
  • B 3 7
  • C 1 7
  • D 9 1 0
  • E 1 1 0

Q19:

Suppose 𝐴 and 𝐵 are events. Given that 𝑃(𝐴)=4𝑥, 𝑃𝐵=𝑥,𝑃(𝐴𝐵)=3𝑥+0.9, and 𝑃(𝐴𝐵)=12𝑥, find the value of 𝑥.

  • A 3 5
  • B 1 4
  • C 1 5
  • D 5 1 9

Q20:

Suppose 𝐴 and 𝐵 are two events in a random experiment. Given that 𝑃(𝐵)=710𝑃(𝐴), 𝑃(𝐴𝐵)=0.12, and 𝑃𝐵𝐴=0.03, find 𝑃(𝐵).

Q21:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃(𝐴)=25 and 𝑃(𝐵)=𝑥. Given that 𝑃(𝐴𝐵)=13 and 𝐴𝐵, find the value of 𝑥.

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

Q22:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃(𝐴𝐵)=0.64 and 𝐴𝐵, find 𝑃(𝐵).

Q23:

Denote by 𝐴 and 𝐵 two independent events. Given that 𝑃(𝐴)=0.54 and 𝑃(𝐵)=0.17, find 𝑃(𝐴𝐵).

Q24:

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?

  • A 1 6 4 9
  • B 3 8 1 1 9
  • C 4 7
  • D 9 4 9

Q25:

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

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