Worksheet: Conditional Probability: Bayes' Theorem

In this worksheet, we will practice computing probabilities using Bayes' rule.

Q1:

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃 ( 𝐴 ) = 0 . 6 3 and 𝑃 ( 𝐵 ) = 0 . 7 7 . Given that 𝑃 ( 𝐵 | 𝐴 ) = 0 . 8 8 , find 𝑃 ( 𝐴 | 𝐵 ) .

Q2:

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that 𝑃 ( 𝐴 ) = 0 . 3 9 and 𝑃 ( 𝐵 | 𝐴 ) = 0 . 8 8 , find 𝑃 𝐵 𝐴 .

Q3:

Matthew rolls two fair dice numbered from one to six and records the results. Let 𝐴 be the event of rolling two numbers whose product is a square number and let 𝐵 be the event of rolling two numbers that are even.

Determine the probability of 𝐴 .

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

Determine the probability of 𝐵 .

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

Determine the probability of ( 𝐴 𝐵 ).

  • A 1 1 2
  • B 2 9
  • C 1 3
  • D 1 1 8
  • E 3 8

Determine the probability of ( 𝐵 𝐴 ) .

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

Is it true that 𝑃 ( 𝐴 ) 𝑃 ( 𝐵 𝐴 ) = 𝑃 ( 𝐴 𝐵 ) and 𝑃 ( 𝐵 ) 𝑃 ( 𝐴 𝐵 ) = 𝑃 ( 𝐴 𝐵 ) ?

  • Ayes
  • Bno

Q4:

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃 ( 𝐴 ) = 0 . 4 5 and 𝑃 ( 𝐵 ) = 0 . 7 . Given that 𝑃 ( 𝐵 | 𝐴 ) = 0 . 8 4 , find 𝑃 ( 𝐴 | 𝐵 ) .

Q5:

Suppose that 𝐴 and 𝐵 are events with probabilities 𝑃 ( 𝐴 ) = 0 . 4 and 𝑃 ( 𝐵 ) = 0 . 6 . Given that 𝑃 ( 𝐵 | 𝐴 ) = 0 . 7 2 , find 𝑃 ( 𝐴 | 𝐵 ) .

Q6:

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that 𝑃 ( 𝐴 ) = 0 . 3 6 and 𝑃 ( 𝐵 | 𝐴 ) = 0 . 7 5 , find 𝑃 𝐵 𝐴 .

Q7:

Suppose that 𝐴 and 𝐵 are events in a random experiment. Given that 𝑃 ( 𝐴 ) = 0 . 4 5 and 𝑃 ( 𝐵 | 𝐴 ) = 0 . 6 6 , find 𝑃 𝐵 𝐴 .

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