smc-2736-20-Set Notation

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-p-p^2`
`p^2-p^3`
`p-p^3`
`1-p + p^3`
`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

`m = n`
`m = 1 - p`
`m + n = 1`
`m = p`
`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 2018 VCAA 14 MC

Two events, `A` and `B`, are independent, where `text(Pr) (B) = 2 text(Pr) (A)` and `text(Pr) (A uu B) = 0.52`

`text(Pr) (A)` is equal to

Show Answers Only

`B`

Show Worked Solution

`text(If)\ \ p_A = text(Pr)(A) \ =>\ 2p_A = text(Pr)(B)`

`text(Pr)(A uu B)`	`= text(Pr)(A) + text(Pr)(B) – text(Pr)(A) xx text(Pr)(B)`
`0.52`	`= p_A + 2p_A – p_A ⋅ 2p_A`
`2(p_A)^2 – 3 p_A + 0.52`	`= 0`
`:. p_A`	`= 0.2`

`=> B`

Probability, MET1 2017 VCAA 8

For events `A` and `B` from a sample space, `text(Pr)(A text(|)B) = 1/5` and `text(Pr)(B text(|)A) = 1/4`. Let `text(Pr)(A nn B) = p`.

Find `text(Pr)(A)` in terms of `p`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
Find `text(Pr)(A^{′} nn B^{′})` in terms of `p`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
Given that `text(Pr)(A uu B) <= 1/5`, state the largest possible interval for `p`. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Pr) (A) = 4p,\ \ p > 0`
`1-8p`
`0 < p <= 1/40`

Show Worked Solution

a.	`text(Pr)(A)`	`=(text(Pr)(A nn B))/(text(Pr)(B text(\|) A))`
		`=p/(1/4)`
		`=4p`

b. `text(Consider the Venn diagram:)`

♦ Mean mark 40%.
MARKER’S COMMENT: The most successful answers used a Venn diagram or table.

`text(Pr)(A^{′} nn B^{′}) = 1-8p`

c. `text(Given Pr)(A uu B) = 8p`

♦ Mean mark 37%.

`=> 0 < 8p <= 1/5`

`:. 0 < p <= 1/40`

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 - p) (1 - q)`
`1 - pq`
`1 - (p + q)`
`2 - p - q`
`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

`text(Pr)(P′) = text(Pr)(Q)`
`text(Pr)(P ∩ Q′) = text(Pr)(P′ ∩ Q)`
`text(Pr)(P ∩ Q) = text(Pr)(P) + Pr(Q)`
`text(Pr)(P ∩ Q′) = text(Pr)(P ∩ Q)`
`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`.

Calculate `text(Pr)(A^{′} ∩ B)` when `text(Pr)(A ∩ B) = 1/8`. (1 mark)

--- 5 WORK AREA LINES (style=lined) ---
Calculate `text(Pr)(A^{′} ∩ B)` when `A` and `B` are mutually exclusive events. (1 mark)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`5/24`
`1/3`

Show Worked Solution

a. `text(Sketch Venn diagram:)`

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

`:. text(Pr)(A^{′} ∩ B)`	`=text(Pr)(B)-text(Pr)(A ∩B)`
	`=1/3-1/8`
	`=5/24`

♦♦ Mean mark (b) 23%.

`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^{′}` denotes the compliment of `A`, calculate `text(Pr) (A^{′} nn B)` when

`text(Pr) (A uu B) = 3/4` (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
`A` and `B` are mutually exclusive. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Pr) (A^{′} nn B) = 3/20`
`text(Pr) (A^{′} nn B) = 1/4`

Show Worked Solution

a. `text(Sketch Venn Diagram)`

`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^{′} nn B) = 1/4-1/10 = 3/20`

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

`:.\ text(Pr) (A^{′} 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

`0`
`1/4`
`3/8`
`1/2`
`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`

Probability, MET2 2013 VCAA 17 MC

`A` and `B` are events of a sample space.

Given that `text(Pr)(A|B) = p,\ \ text(Pr)(B) = p^2` and `text(Pr)(A) = p^(1/3),\ text(Pr)(B|A)` is equal to

`p`
`p^(4/3)`
`p^(7/3)`
`p^(8/3)`
`p^3`

Show Answers Only

`D`

Show Worked Solution

`text(Pr)(A \| B)`	`= (text(Pr)(A ∩ B))/(text(Pr)(B)`
`p`	`= (text(Pr)(A ∩ B))/(p^2)`
`:. text(Pr)(A ∩ B)`	`= p^3`

♦ Mean mark 49%.

`text(Pr)(B \| A)`	`= (text(Pr)(A ∩ B))/(text(Pr)(A))`
	`= (p^3)/(p^(1/3))`
`:. text(Pr)(B \| A)`	`= p^(8/3)`

`=> D`

Probability, MET2 2012 VCAA 13 MC

`A` and `B` are events of a sample space `S.`

`text(Pr)(A nn B) = 2/5` and `text(Pr)(A nn B prime) = 3/7`

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

`6/35`
`15/29`
`14/35`
`29/35`
`2/3`

Show Answers Only

`B`

Show Worked Solution

`text(Pr)(B prime \| A)`	`= (text(Pr)(B prime nn A))/(text(Pr)(A))`
	`= (text(Pr)(B prime nn A))/(text(Pr)(B prime nn A)+ text(Pr)(A nn B))`
	`= (3/7)/(3/7 + 2/5)`
	`= 15/29`

`=> B`