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Probability, MET1 2015 VCAA 8

For events `A` and `B` from a sample space, `text(Pr)(A | B) = 3/4`  and  `text(Pr)(B) = 1/3`.

  1. Calculate  `text(Pr)(A ∩ B)`.   (1 mark)

  2. Calculate  `text(Pr)(A^{′} ∩ B)`, where `A^{′}` denotes the complement of `A`.   (1 mark)

  3. If events `A` and `B` are independent, calculate  `text(Pr)(A ∪ B)`.   (1 mark)

Show Answers Only

  1. `1/4`
  2. `1/12`
  3. `5/6`

Show Worked Solution

a.   `text(Using Conditional Probability:)`

`text(Pr)(A | B)` `= (text(Pr)(A ∩ B))/(text(Pr)(B))`
`3/4` `= (text(Pr)(A ∩ B))/(1/3)`
`:. text(Pr)(A ∩ B)` `= 1/4`

 

b.    met1-2015-vcaa-q8-answer
`text(Pr)(A^{′} ∩ B)` `= text(Pr)(B)-text(Pr)(A ∩B)`
  `= 1/3-1/4`
  `= 1/12`

 

c.   `text(If)\ A, B\ text(independent)`

♦♦ Mean mark 28%.
MARKER’S COMMENT: A lack of understanding of independent events was clearly evident.

`text(Pr)(A ∩ B)` `= text(Pr)(A) xx Pr(B)`
`1/4` `= text(Pr)(A) xx 1/3`
`:. text(Pr)(A)` `= 3/4`

 

`text(Pr)(A ∪ B)` `= text(Pr)(A) + text(Pr)(B)-text(Pr)(A ∩ B)`
  `= 3/4 + 1/3-1/4`
`:. text(Pr)(A ∪ B)` `= 5/6`