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Worksheet: Unions, Intersections, and Complements

Q1:

Suppose is an event such that . Determine .

  • A0
  • B
  • C1
  • D

Q2:

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?

Q3:

Suppose 𝐴 , 𝐡 , and 𝐢 are three mutually exclusive events in a sample space 𝑆 . Given that 𝑆 = 𝐴 βˆͺ 𝐡 βˆͺ 𝐢 , 𝑃 ( 𝐴 ) = 6 𝑃 ( 𝐡 ) , and 𝑃 ( 𝐢 ) = 1 6 𝑃 ( 𝐴 ) , find 𝑃 ( 𝐴 βˆͺ 𝐢 ) .

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

Q4:

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

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

Q5:

Suppose and are two events with probabilities and . Given that , determine .

Q6:

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

Q7:

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

Q8:

Suppose 𝐴 and 𝐡 are two events. Given that 𝐡 βŠ‚ 𝐴 , 𝑃 ( 𝐡 ) = 4 9 , and 𝑃 ( 𝐴 βˆ’ 𝐡 ) = 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

Q9:

Suppose that and are two events with probabilities and . Given that , determine .

  • A
  • B
  • C
  • D
  • E

Q10:

Suppose and are two events with probabilities and . Given that , determine the probability that at least one of the events and occurs.

Q11:

Suppose and are two events with probabilities and . Given that , determine the probability that only one of the events and occurs.

Q12:

A ball is drawn at random from a bag containing 94 balls each with a unique number from 1 to 94. Suppose 𝐴 is the event of drawing an odd number and 𝐡 is the event of drawing a prime number. Find 𝑃 ( 𝐴 βˆ’ 𝐡 ) .

  • A 1 9 4
  • B 2 3 9 4
  • C 1 2
  • D 1 2 4 7

Q13:

Suppose 𝐴 and 𝐡 are events in the sample space of an experiment. Given that 𝑃 ( 𝐴 β€² ) = 5 8 and 𝑃 ( 𝐴 β€² ∩ 𝐡 β€² ) = 3 8 , find the value of 𝑃 ( 𝐴 β€² βˆ’ 𝐡 β€² ) .

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

Q14:

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

  • A 𝐡
  • B 𝐴
  • C 𝐴 βˆ’ 𝐡
  • D βˆ…

Q15:

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

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

Q16:

Suppose 𝐴 and 𝐡 are events such that 𝐡 βŠ‚ 𝐴 . Determine 𝐴 βˆͺ 𝐡 .

  • A βˆ…
  • B 𝐡
  • C 𝐴 ∩ 𝐡
  • D 𝐴

Q17:

Suppose and are two events. Given that , , and , find .

  • A
  • B
  • C
  • D

Q18:

Suppose that 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐡 ) = 5 8 , 𝑃 ( 𝐴 βˆͺ 𝐡 ) = 3 4 , and 𝐡 βŠ‚ 𝐴 , find 𝑃 ( 𝐴 ) .

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

Q19:

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?

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

Q20:

Suppose and are events. Given that , , , and , find the value of .

  • A
  • B
  • C
  • D

Q21:

Suppose 𝐴 and 𝐡 are events. Given that 𝑃 ( 𝐴 ) = 4 π‘₯ , 𝑃 ο€Ί 𝐡  = π‘₯ , 𝑃 ( 𝐴 βˆͺ 𝐡 ) = 3 π‘₯ + 0 . 9 , and 𝑃 ( 𝐴 ∩ 𝐡 ) = 1 2 π‘₯ , find the value of π‘₯ .

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

Q22:

Suppose and are two events in a random experiment. Given that , , and , find .

Q23:

Suppose and are two events with probabilities and . Given that and , find the value of .

  • A
  • B
  • C
  • D

Q24:

Suppose 𝐴 and 𝐡 are two events. Given that 𝑃 ( 𝐴 βˆͺ 𝐡 ) = 0 . 6 4 and 𝐴 βŠ‚ 𝐡 , find 𝑃 ( 𝐡 ) .

Q25:

Suppose and are two events. Given that , , and , find .