Worksheet: Probability of Unions, Intersections, and Complements

In this worksheet, we will practice using unions, intersections, and complements of events in probability calculations.

Q1:

Suppose 𝐴 is an event such that 𝑃 ( 𝐴 ) = 4 5 . Determine 𝑃 𝐴 .

  • A0
  • B 4 5
  • C1
  • D 1 5

Q2:

Denote by 𝐴 and 𝐵 two events with probabilities 𝑃 ( 𝐴 ) = 0 . 2 and 𝑃 ( 𝐵 ) = 0 . 4 7 . Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 1 8 , find 𝑃 ( 𝐴 𝐵 ) .

Q3:

Denote by 𝐴 and 𝐵 two events with probabilities 𝑃 ( 𝐴 ) = 0 . 5 8 and 𝑃 ( 𝐵 ) = 0 . 2 . Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 6 4 , find 𝑃 ( 𝐴 𝐵 ) .

Q4:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃 ( 𝐴 ) = 0 . 8 and 𝑃 ( 𝐵 ) = 0 . 8 . Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 1 , determine the probability that at least one of the events 𝐴 and 𝐵 occurs.

Q5:

Denote by 𝐴 and 𝐵 two events in a sample space. Given that 𝑃 ( 𝐴 ) = 0 . 4 5 , 𝑃 ( 𝐵 ) = 0 . 6 , and 𝑃 ( 𝐴 𝐵 ) = 0 . 7 8 , are the events 𝐴 and 𝐵 independent?

  • Ayes
  • Bno

Q6:

Suppose 𝐴 and 𝐵 are two events with probability 𝑃 ( 𝐴 ) = 0 . 6 and 𝑃 ( 𝐵 ) = 0 . 5 . Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 4 , what is the probability that at least one of the events does not occur?

Q7:

Suppose 𝐴 , 𝐵 , and 𝐶 are three mutually exclusive events in a sample space 𝑆 . Given that 𝑆 = 𝐴 𝐵 𝐶 , 𝑃 ( 𝐴 ) = 1 5 𝑃 ( 𝐵 ) , and 𝑃 ( 𝐶 ) = 4 𝑃 ( 𝐴 ) , find 𝑃 ( 𝐵 𝐶 ) .

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

Q8:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃 ( 𝐴 ) = 5 7 and 𝑃 ( 𝐵 ) = 4 7 . Given that 𝑃 ( 𝐴 𝐵 ) = 6 7 , determine 𝑃 ( 𝐴 𝐵 ) .

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

Q9:

Suppose 𝑋 and 𝑌 are two events with probabilities 𝑃 ( 𝑋 ) = 0 . 4 9 and 𝑃 ( 𝑌 ) = 0 . 4 8 . Given that 𝑃 ( 𝑋 𝑌 ) = 0 . 9 5 , determine 𝑃 ( 𝑋 𝑌 ) .

Q10:

Suppose 𝐴 and 𝐵 are two events. Given that 𝐵 𝐴 , 𝑃 ( 𝐵 ) = 4 9 , and 𝑃 ( 𝐴 𝐵 ) = 1 5 , determine 𝑃 ( 𝐴 ) .

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

Q11:

Suppose that 𝑋 and 𝑌 are two events with probabilities 𝑃 ( 𝑌 ) = 1 3 and 𝑃 ( 𝑋 ) = 𝑃 𝑋 . Given that 𝑃 ( 𝑋 𝑌 ) = 1 8 , determine 𝑃 ( 𝑋 𝑌 ) .

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

Q12:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃 ( 𝐴 ) = 0 . 6 and 𝑃 ( 𝐵 ) = 0 . 5 . Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 3 , determine the probability that only one of the events 𝐴 and 𝐵 occurs.

Q13:

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 ( 𝐴 𝐵 ) .

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

Q14:

Suppose 𝐴 and 𝐵 are events in the sample space of an experiment. Given that 𝑃 ( 𝐴 ) = 1 2 and 𝑃 ( 𝐴 𝐵 ) = 1 1 2 , find the value of 𝑃 ( 𝐴 𝐵 ) .

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

Q15:

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

  • A 𝐵
  • B 𝐴
  • C 𝐴 𝐵
  • D

Q16:

𝐴 and 𝐵 are two events in a sample space of a random experiment where 𝑃 ( 𝐴 ) = 3 1 0 , 𝑃 ( 𝐵 ) = 1 5 , and 𝑃 ( 𝐴 𝐵 ) = 1 1 0 . Find 𝑃 ( 𝐴 𝐵 ) .

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

Q17:

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

  • A
  • B 𝐵
  • C 𝐴 𝐵
  • D 𝐴

Q18:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃 ( 𝐴 ) = 5 8 , 𝑃 ( 𝐵 ) = 3 4 , and 𝑃 ( 𝐴 𝐵 ) = 1 4 , find 𝑃 𝐴 𝐵 .

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

Q19:

Suppose that 𝐴 and 𝐵 are two events. Given that 𝑃 ( 𝐵 ) = 5 8 , 𝑃 ( 𝐴 𝐵 ) = 3 4 , and 𝐵 𝐴 , find 𝑃 ( 𝐴 ) .

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

Q20:

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

Q21:

Suppose 𝐴 and 𝐵 are events. Given that 𝐴 𝐵 , 𝑃 ( 𝐴 ) = 𝑥 , 𝑃 𝐵 = 7 𝑥 , and 𝑃 ( 𝐴 𝐵 ) = 7 𝑥 + 0 . 4 , find the value of 𝑥 .

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

Q22:

Suppose 𝐴 and 𝐵 are events. Given that 𝑃 ( 𝐴 ) = 4 𝑥 , 𝑃 𝐵 = 𝑥 , 𝑃 ( 𝐴 𝐵 ) = 3 𝑥 + 0 . 9 , and 𝑃 ( 𝐴 𝐵 ) = 1 2 𝑥 , find the value of 𝑥 .

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

Q23:

Suppose 𝐴 and 𝐵 are two events in a random experiment. Given that 𝑃 ( 𝐵 ) = 7 1 0 𝑃 ( 𝐴 ) , 𝑃 ( 𝐴 𝐵 ) = 0 . 1 2 , and 𝑃 𝐵 𝐴 = 0 . 0 3 , find 𝑃 ( 𝐵 ) .

Q24:

Suppose 𝐴 and 𝐵 are two events with probabilities 𝑃 ( 𝐴 ) = 2 5 and 𝑃 ( 𝐵 ) = 𝑥 . Given that 𝑃 ( 𝐴 𝐵 ) = 1 3 and 𝐴 𝐵 , find the value of 𝑥 .

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

Q25:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃 ( 𝐴 𝐵 ) = 0 . 6 4 and 𝐴 𝐵 , find 𝑃 ( 𝐵 ) .

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