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`
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`
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, 2, 3} and {1, 2}`
- `{1, 2} and {3, 4}`
- `{1, 3, 5} and {1, 4, 6}`
- `{1, 2} and {1, 3, 4, 6}`
- `{1, 2} and {2, 4, 6}`
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, 3, 5, 7, 9, 11} and {1, 4, 7, 10}`
- `{1, 3, 5, 7, 9, 11} and {2, 4, 6, 8, 10, 12}`
- `{4, 8, 12} and {6, 12}`
- `{6, 12} and {1, 12}`
- `{2, 4, 6, 8, 10, 12} and {1, 2, 3}`
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 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 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`. --- 5 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
a. `text(Sketch Venn diagram:)` `text(Mutually exclusive means)\ \ text(Pr)(A ∩ B)=0,` `:. text(Pr)(A^{′} ∩ B) = 1/3`Show Worked Solution
♦♦ 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%.
b.
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 --- 5 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
a. `text(Sketch Venn Diagram)` `:.\ 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`Show Worked Solution
`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`
b.
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`