# Thread: Simple beginners probability question

1. ## Simple beginners probability question

Hi, could you check over my work to see if this is correct?

You are given $\displaystyle P(A \cup B)=0.7$ and $\displaystyle P(A \cup B\prime)=0.9$. Determine P(A).

Find B:
$\displaystyle P(A \cup B\prime)=1-B=0.9 \\ B=0.1$

Then:
$\displaystyle P(A)=P(A\cup B)-P(B)=0.7-0.1 \\ =0.6$

2. Originally Posted by greatsheelephant
Hi, could you check over my work to see if this is correct?

You are given $\displaystyle P(A \cup B)=0.7$ and $\displaystyle P(A \cup B\prime)=0.9$. Determine P(A).

Find B:
$\displaystyle P(A \cup B\prime)=1-B=0.9 \\ B=0.1$

Then:
$\displaystyle P(A)=P(A\cup B)-P(B)=0.7-0.1 \\ =0.6$
$\displaystyle P(A)=P(A\cup B)-P(B)+P(A\cap B)$

3. From the given we get:
$\displaystyle P(A) + P(B) - P(A \cap B) = 0.7~\&$
$\displaystyle P(A) + P(B') - P(A \cap B') = 0.9$.
Use these facts: $\displaystyle P(B)+P(B')=1~\&~ P(A \cap B) + P(A \cap B')=P(A)$.

4. Hello, greatsheelephant!

$\displaystyle \text{Given: }\;P(A \cup B)\,=\,0.7,\;P(A \cup B')\,=\,0.9$
$\displaystyle \text{Determine }P(A).$

Using DeMorgan's Laws: .$\displaystyle \begin{Bmatrix} P(A \cup B) \:=\:0.7 & \Rightarrow & P(A' \cap B') \:=\: 0.3 \\ \\[-3mm] P(A \cup B') \:=\: 0.9 & \Rightarrow & P(A' \cap B) \:=\: 0.1 \end{Bmatrix}$

I used a Venn diagram:

Code:

*-------------------------------*
|                               |
|   *---------------*           |
|   |/A/////////////|           |
|   |///////////////|           |
|   |///////*-------+-------*   |
|   |///////|///////|       |   |
|   |///////|///////|       |   |
|   |///////|///////|  0.1  |   |
|   *-------+-------*       |   |
|           |             B |   |
|   0.3     *---------------*   |
|                               |
*-------------------------------*

Then it is evident that: .$\displaystyle P(A) \,=\,0.6$

5. Thank you! I came to 0.6 as well but I went about it by finding B, then subtracting the remaining union from the sample space, 1.