To find the probability [tex]\(\operatorname{Pr}\left(A \cup B^{\prime}\right)\)[/tex], we will use the following steps and relationships in probability theory:
### Step 1: Understand key probabilities
- [tex]\(\operatorname{Pr}(A) = 0.3\)[/tex]
- [tex]\(\operatorname{Pr}(B) = 0.4\)[/tex]
- [tex]\(\operatorname{Pr}\left(A \cap B^{\prime}\right) = 0.2\)[/tex]
### Step 2: Compute [tex]\(\operatorname{Pr}(B^{\prime})\)[/tex]
[tex]\[
\operatorname{Pr}(B^{\prime}) = 1 - \operatorname{Pr}(B)
\][/tex]
Since [tex]\(\operatorname{Pr}(B) = 0.4\)[/tex], we have:
[tex]\[
\operatorname{Pr}(B^{\prime}) = 1 - 0.4 = 0.6
\][/tex]
### Step 3: Use the formula for union of events
To find [tex]\(\operatorname{Pr}\left(A \cup B^{\prime}\right)\)[/tex], use the formula:
[tex]\[
\operatorname{Pr}\left(A \cup B^{\prime}\right) = \operatorname{Pr}(A) + \operatorname{Pr}(B^{\prime}) - \operatorname{Pr}\left(A \cap B^{\prime}\right)
\][/tex]
### Step 4: Substitute the known values into the formula
We know:
- [tex]\(\operatorname{Pr}(A) = 0.3\)[/tex]
- [tex]\(\operatorname{Pr}(B^{\prime}) = 0.6\)[/tex]
- [tex]\(\operatorname{Pr}\left(A \cap B^{\prime}\right) = 0.2\)[/tex]
So, substitute these into the formula:
[tex]\[
\operatorname{Pr}\left(A \cup B^{\prime}\right) = 0.3 + 0.6 - 0.2
\][/tex]
### Step 5: Perform the arithmetic
[tex]\[
\operatorname{Pr}\left(A \cup B^{\prime}\right) = 0.3 + 0.6 - 0.2 = 0.7
\][/tex]
Hence, the probability [tex]\(\operatorname{Pr}\left(A \cup B^{\prime}\right)\)[/tex] is [tex]\(0.7\)[/tex].
