To find [tex]\( P(A \text{ or } B) \)[/tex], we use the formula for the union of two events in probability. The formula is:
[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 A occurring.
- [tex]\( P(B) \)[/tex] is the probability of event B occurring.
- [tex]\( P(A \text{ and } B) \)[/tex] is the probability of both events A and B occurring simultaneously.
Given:
- [tex]\( P(A) = 0.50 \)[/tex]
- [tex]\( P(B) = 0.30 \)[/tex]
- [tex]\( P(A \text{ and } B) = 0.15 \)[/tex]
We substitute these values into the formula:
[tex]\[ P(A \text{ or } B) = 0.50 + 0.30 - 0.15 \][/tex]
Now, we perform the arithmetic step-by-step:
1. Add [tex]\( P(A) \)[/tex] and [tex]\( P(B) \)[/tex]:
[tex]\[ 0.50 + 0.30 = 0.80 \][/tex]
2. Subtract [tex]\( P(A \text{ and } B) \)[/tex]:
[tex]\[ 0.80 - 0.15 = 0.65 \][/tex]
Therefore, the probability of either event A or event B occurring is [tex]\( P(A \text{ or } B) = 0.65 \)[/tex].
So, the correct answer is:
B. 0.65
