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If [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are events with [tex]\( P(A) = 0.3 \)[/tex], [tex]\( P(A \text{ OR } B) = 0.63 \)[/tex], and [tex]\( P(A \text{ AND } B) = 0.17 \)[/tex], find [tex]\( P(B) \)[/tex].

Provide your answer below:
[tex]\(\square\)[/tex]

Sagot :

GinnyAnswer
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]
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