To determine the probability of either event [tex]\(A\)[/tex] or event [tex]\(B\)[/tex] occurring, we use the formula for the probability of the union of two events. This formula is given by:
[tex]\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\][/tex]
Where:
- [tex]\(P(A)\)[/tex] is the probability of event [tex]\(A\)[/tex] occurring.
- [tex]\(P(B)\)[/tex] is the probability of event [tex]\(B\)[/tex] occurring.
- [tex]\(P(A \text{ and } B)\)[/tex] is the probability that both events [tex]\(A\)[/tex] and [tex]\(B\)[/tex] occur simultaneously.
Given:
- [tex]\(P(A) = 0.60\)[/tex]
- [tex]\(P(B) = 0.20\)[/tex]
- [tex]\(P(A \text{ and } B) = 0.15\)[/tex]
Substitute these values into the formula:
[tex]\[
P(A \text{ or } B) = 0.60 + 0.20 - 0.15
\][/tex]
First, add [tex]\(P(A)\)[/tex] and [tex]\(P(B)\)[/tex]:
[tex]\[
0.60 + 0.20 = 0.80
\][/tex]
Next, subtract [tex]\(P(A \text{ and } B)\)[/tex] from the result:
[tex]\[
0.80 - 0.15 = 0.65
\][/tex]
Therefore, the probability of either event [tex]\(A\)[/tex] or event [tex]\(B\)[/tex] occurring is:
[tex]\[
P(A \text{ or } B) = 0.65
\][/tex]
The correct answer is:
C. 0.65
