To find the probability [tex]\( P(B) \)[/tex], we can use the formula for the probability of the union of two events:
[tex]\[ P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \][/tex]
We are given the following probabilities:
- [tex]\( P(A) = 0.3 \)[/tex]
- [tex]\( P(A \text{ OR } B) = 0.63 \)[/tex]
- [tex]\( P(A \text{ AND } B) = 0.17 \)[/tex]
We need to isolate [tex]\( P(B) \)[/tex] in the formula. We can rearrange the formula to solve for [tex]\( P(B) \)[/tex]:
[tex]\[
P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B)
\][/tex]
Rearranging to solve for [tex]\( P(B) \)[/tex]:
[tex]\[
P(B) = P(A \text{ OR } B) - P(A) + P(A \text{ AND } B)
\][/tex]
By substituting the known values into the equation:
[tex]\[
P(B) = 0.63 - 0.3 + 0.17
\][/tex]
Performing the arithmetic operations step-by-step:
[tex]\[
P(B) = 0.63 - 0.3 = 0.33
\][/tex]
[tex]\[
P(B) = 0.33 + 0.17 = 0.5
\][/tex]
Therefore, the probability [tex]\( P(B) \)[/tex] is:
[tex]\[
P(B) = 0.5
\][/tex]
