Certainly! Let's break down the calculation step-by-step.
We need to find the probability of issue [tex]$B$[/tex] occurring, denoted as [tex]$P(B)$[/tex].
We know:
- The probability of issue [tex]$A$[/tex] occurring, [tex]\( P(A) = 0.1 \)[/tex].
- The probability of either issue [tex]$A$[/tex] or issue [tex]$B$[/tex] occurring (or both), [tex]\( P(A \text{ OR } B) = 0.93 \)[/tex].
- The probability of both issues [tex]$A$[/tex] and [tex]$B$[/tex] occurring together, [tex]\( P(A \text{ AND } B) = 0.07 \)[/tex].
We can use the formula for the union of two events to find [tex]\( P(B) \)[/tex]:
[tex]\[ P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \][/tex]
Now we will rearrange this formula to solve for [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = P(A \text{ OR } B) - P(A) + P(A \text{ AND } B)
\][/tex]
Substituting in the known values:
[tex]\[
P(B) = 0.93 - 0.1 + 0.07
\][/tex]
Then, we perform the subtraction and addition:
[tex]\[
P(B) = 0.93 - 0.1 = 0.83
\][/tex]
[tex]\[
P(B) = 0.83 + 0.07 = 0.90
\][/tex]
Therefore, the probability of issue [tex]$B$[/tex] occurring is:
[tex]\[
P(B) = 0.90
\][/tex]
