To find [tex]\( P(B) \)[/tex] given the probabilities of the events [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we can use the key probability formulas. Here's the step-by-step process:
1. List the given probabilities:
- [tex]\( P(A) = 0.5 \)[/tex]
- [tex]\( P(A \text{ OR } B) = 0.65 \)[/tex]
- [tex]\( P(A \text{ AND } B) = 0.15 \)[/tex]
2. Recall the formula for the union of two events:
[tex]\[
P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B)
\][/tex]
3. Rearrange the formula to solve for [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = P(A \text{ OR } B) - P(A) + P(A \text{ AND } B)
\][/tex]
4. Substitute the given values into the formula:
[tex]\[
P(B) = 0.65 - 0.5 + 0.15
\][/tex]
5. Perform the arithmetic operations:
[tex]\[
P(B) = 0.65 - 0.5 + 0.15 = 0.15 + 0.15 = 0.3
\][/tex]
Therefore, the probability of event [tex]\( B \)[/tex], [tex]\( P(B) \)[/tex], is [tex]\( 0.3 \)[/tex].
