Lesson: Probability of Unions, Intersections, and Complements

In this lesson, we will learn how to use unions, intersections, and complements of events in probability calculations.

Sample Question Videos

  • 01:22

Worksheet: Probability of Unions, Intersections, and Complements • 25 Questions • 1 Video

Q1:

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

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?

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 ) = 6 𝑃 ( 𝐵 ) , and 𝑃 ( 𝐶 ) = 1 6 𝑃 ( 𝐴 ) , find 𝑃 ( 𝐴 ∪ 𝐶 ) .

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 ) .

Q9:

Suppose 𝑋 and 𝑌 are two events with probabilities 𝑃 ( 𝑋 ) = 0 . 6 2 and 𝑃 ( 𝑌 ) = 0 . 1 8 . Given that 𝑃 ( 𝑋 ∪ 𝑌 ) = 0 . 7 1 , 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 ) .

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 ) .

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 94 balls each with a unique number from 1 to 94. Suppose 𝐴 is the event of drawing an odd number and 𝐵 is the event of drawing a prime number. Find 𝑃 ( 𝐴 − 𝐵 ) .

Q14:

Suppose 𝐴 and 𝐵 are events in the sample space of an experiment. Given that 𝑃 ( 𝐴 ′ ) = 5 8 and 𝑃 ( 𝐴 ′ ∩ 𝐵 ′ ) = 3 8 , find the value of 𝑃 ( 𝐴 ′ − 𝐵 ′ ) .

Q15:

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

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 ) .

Q17:

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

Q18:

Suppose 𝐴 and 𝐵 are two events. Given that 𝑃 ( 𝐴 ) = 1 2 , 𝑃 ( 𝐵 ) = 2 3 , and 𝑃 ( 𝐴 − 𝐵 ) = 1 3 , find 𝑃 ( 𝐴 ∪ 𝐵 ) ′ .

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 ) .

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?

Q21:

Suppose 𝐴 and 𝐵 are events. Given that 𝐴 ⊂ 𝐵 , 𝑃 ( 𝐴 ) = 𝑥 , 𝑃 ( 𝐵 ) = 9 𝑥 ′ , and 𝑃 ( 𝐴 ∪ 𝐵 ) = 8 𝑥 + 0 . 7 , find the value of 𝑥 .

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 .

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 ) = 1 8 and 𝑃 ( 𝐵 ) = 𝑥 . Given that 𝑃 ( 𝐴 ∪ 𝐵 ) = 1 4 ′ and 𝐴 ⊂ 𝐵 , find the value of 𝑥 .

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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