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Probability, MET2 2021 VCAA 20 MC

Let `A` and `B` be two independent events from a sample space.

If  `text{Pr} (A) = p , \ text{Pr}(B) = p^2`  and  `text{Pr} (A) + text{Pr} (B) = 1`, then  `text{Pr}(A′ ∪ B)`  is equal to

  1. `1 - p - p^2`
  2. `p^2 - p^3`
  3. `p - p^3`
  4. `1 - p + p^3`
  5. `1 - p - p^2 + p^3`
Show Answers Only

`D`

Show Worked Solution

`text{Pr} (A) = p , \ text{Pr}(B) = p^2, text{Pr} (A) + text{Pr} (B) = 1`

♦ Mean mark 39%.
`text{Pr}(A ∩ B)` `= p xx p^2 = p^3 \ \ (A, B\ text{are independent)}`
`text{Pr}(A ∩ B′)` `= p (1 – p^2) = p – p^3`
`text{Pr}(A′ ∪ B)` `= 1 – text{Pr} (A ∩ B′)`
  `= 1 – p + p^3`

 
`=> D`

Probability, MET2 2019 VCAA 11 MC

`A` and `B` are events from a sample space such that  `text(Pr)(A) = p`, where  `p > 0, \ text(Pr)(B\ text{|}\ A) = m`  and  `text(Pr)(B\ text{|}\ A prime) = n`.

`A` and `B` are independent events when

  1. `m = n`
  2. `m = 1 - p`
  3. `m + n = 1`
  4. `m = p`
  5. `m + n = 1 - p`
Show Answers Only

`A`

Show Worked Solution

`text(S) text(ince)\ A and B\ text(are independent),`

`text(Pr)(B\ text{|}\ A) = text(Pr)(B\ text{|}\ A prime)`

`:. m = n`
 

`=>   A`

Probability, MET2 2008 VCAA 15 MC

The sample space when a fair die is rolled is `{1, 2, 3, 4, 5, 6}`, with each outcome being equally likely.

For which of the following pairs of events are the events independent?

  1. `{1, 2, 3} and {1, 2}`
  2. `{1, 2} and {3, 4}`
  3. `{1, 3, 5} and {1, 4, 6}`
  4. `{1, 2} and {1, 3, 4, 6}`
  5. `{1, 2} and {2, 4, 6}`
Show Answers Only

`E`

Show Worked Solution
`text(Let)\ \ A` `= {1, 2}`
`B` `= {2, 4, 6}`
`A nn B` `= {2}`
♦♦♦ Mean mark 7%!
MARKER’S COMMENT: Almost two-thirds of students chose option B, which contained mutually exclusive events.
`text(Pr)(A)` `= 1/3`
`text(Pr)(B)` `= 1/2`
`text(Pr)(A nn B)` `= 1/6`

 

`text(S) text(ince)\ \ text(Pr) (A) xx text(Pr) (B)` `= text(Pr)(A nn B)`
`1/3 xx 1/2` `= 1/6`

 

`:. A, B\ \ text(are independent.)`

`=>   E`

Probability, MET2 2009 VCAA 17 MC

The sample space when a fair twelve-sided die is rolled is `{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}`. Each outcome is equally likely.

For which one of the following pairs of events are the events independent?

  1. `{1, 3, 5, 7, 9, 11} and {1, 4, 7, 10}`
  2. `{1, 3, 5, 7, 9, 11} and {2, 4, 6, 8, 10, 12}`
  3. `{4, 8, 12} and {6, 12}`
  4. `{6, 12} and {1, 12}`
  5. `{2, 4, 6, 8, 10, 12} and {1, 2, 3}`
Show Answers Only

`A`

Show Worked Solution

`text(Consider option A:)`

♦♦♦ Mean mark 31%.
MARKER’S COMMENT: Many chose option B, which contains mutually exclusive events, for which `text(Pr) (A nn B)=0`

`text(Let)\ \A = {1, 3, 5, 7, 9, 11}`

`text(Let)\ \ B = {1, 4, 7, 10}`

`A nn B = {1, 7}`
 

`text(If independent events,)`

`text(Pr) (A nn B) = text(Pr) (A) xx text(Pr) (B)`

`text(Pr) (A) = 6/12=1/2`

`text(Pr) (B) = 4/12=1/3`

`text(Pr) (A nn B) = 2/12=1/6`

 

`:.\ text(Pr) (A nn B) = text(Pr) (A) xx text(Pr) (B)`

`:. A, B\ \ text(are independent)`

`=>   A`

Probability, MET2 2010 VCAA 21 MC

Events `A` and `B` are mutually exclusive events of a sample space with

`text(Pr) (A) = p and text(Pr) (B) = q\ \ text(where)\ \ 0 < p < 1 and 0 < q < 1.`

`text(Pr) (A prime nn B prime)` is equal to

  1. `(1 - p) (1 - q)`
  2. `1 - pq`
  3. `1 - (p + q)`
  4. `2 - p - q`
  5. `1 - (p + q - pq)`
Show Answers Only

`C`

Show Worked Solution

♦ Mean mark 43%.
`text(Pr) (A prime nn B prime)` `= 1 – p – q`
  `= 1 – (p + q)`

`=>   C`

Probability, MET2 2011 VCAA 21 MC

For two events, `P` and `Q`, `text(Pr)(P ∩ Q) = text(Pr)(P′ ∩ Q)`.

`P` and `Q` will be independent events exactly when

  1. `text(Pr)(P′) = text(Pr)(Q)`
  2. `text(Pr)(P ∩ Q′) = text(Pr)(P′ ∩ Q)`
  3. `text(Pr)(P ∩ Q) = text(Pr)(P) + Pr(Q)`
  4. `text(Pr)(P ∩ Q′) = text(Pr)(P ∩ Q)`
  5. `text(Pr)(P) = 1/2`
Show Answers Only

`=> E`

Show Worked Solution

`text(Let)\ \ text(Pr)(P ∩ Q)` `= x = text(Pr)(P′ ∩ Q)`
`text(Let)\ \ text(Pr)(P ∩ Q′)` `= y`

 

`text(If)\ P, Q\ text(independent)`

♦♦♦ Mean mark 15%.
`text(Pr)(P) xx text(Pr)(Q)` `= text(Pr)(P ∩ Q)`
`(y + x)(2x)` `= x`
`:. 2(x + y)` `= 1`
`x + y` `= 1/2`
`text(Pr)(P)` `= 1/2`

`=> E`

Probability, MET1 2007 VCAA 6

Two events, `A` and `B`, from a given event space, are such that  `text(Pr)(A) = 1/5`  and  `text(Pr)(B) = 1/3`.

  1. Calculate  `text(Pr)(A′ ∩ B)`  when  `text(Pr)(A ∩ B) = 1/8`.  (1 mark)
  2. Calculate  `text(Pr)(A′ ∩ B)`  when `A` and `B` are mutually exclusive events.  (1 mark)
Show Answers Only
  1. `5/24`
  2. `1/3`
Show Worked Solution

a.   `text(Sketch Venn diagram:)`

♦♦ Mean mark 31%.
MARKER’S COMMENTS: Students who drew a Venn diagram or Karnaugh map were the most successful.

met1-2007-vcaa-q6-answer3

`:. text(Pr)(A′ ∩ B)` `=text(Pr)(B) – text(Pr)(A ∩B)`
  `=1/3 – 1/8`
  `=5/24`

 

♦♦ Mean mark 23%.
b.    met1-2007-vcaa-q6-answer4

`text(Mutually exclusive means)\ \ text(Pr)(A ∩ B)=0,`

`:. text(Pr)(A′ ∩ B) = 1/3`

Probability, MET1 2011 VCAA 8

Two events, `A` and `B`, are such that  `text(Pr) (A) = 3/5`  and  `text(Pr) (B) = 1/4.`

If `A prime` denotes the compliment of `A`, calculate  `text(Pr) (A prime nn B)` when

  1. `text(Pr) (A uu B) = 3/4`  (2 marks)
  2. `A` and `B` are mutually exclusive.  (1 mark)
Show Answers Only
  1. `text(Pr) (A prime nn B) = 3/20`
  2. `text(Pr) (A prime nn B) = 1/4`
Show Worked Solution

a.   `text(Sketch Venn Diagram)`

vcaa-2011-meth-8i

`text(Pr) (A uu B)` `= text(Pr) (A) + text(Pr) (B) – text(Pr) (A nn B)`
`3/4` `= 3/5 + 1/4 – text(Pr) (A nn B)`
`text(Pr) (A nn B)` `= 1/10`

 

 `:.\ text(Pr) (A prime nn B) = 1/4 – 1/10 = 3/20`

 

b.   vcaa-2011-meth-8ii

`text(Pr) (A∩ B)=0\ \ text{(mutually exclusive)},`

`:.\ text(Pr) (A prime nn B) = text(Pr) (B) = 1/4`

Probability, MET2 2013 VCAA 10 MC

For events `A` and `B,\ text(Pr)(A ∩ B) = p,\ text(Pr)(A′∩ B) = p - 1/8`  and  `text(Pr)(A ∩ B prime) = (3p)/5.`

If `A` and `B` are independent, then the value of  `p`  is

  1. `0`
  2. `1/4`
  3. `3/8`
  4. `1/2`
  5. `3/5`
Show Answers Only

`C`

Show Worked Solution
`text{Pr}(A)` `= text{Pr}(A ∩ B) + text{Pr}(A ∩ B prime)`
  `= p + (3p)/5`
  `= (8p)/5`

 

`text{Pr}(B)` `= text{Pr}(B ∩ A) + text{Pr}(B ∩ A prime)`
  `= p + p – 1/8`
  `= 2p – 1/8`

 

`text(S)text(ince)\ A and B\ text(are independent events,)`

`text{Pr}(A ∩ B)` ` = text{Pr}(A) xx text{Pr}(B)`
`p` `=(8p)/5 (2p – 1/8)`
`5p` `=16p^2-p`
`16p^2-6p` `=0`
`2p(8p-3)` `=0`
`:.p` `=3/8,\ \ \ p!=0`

`=>   C`