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If [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are events with [tex]\( P(A) = 0.9 \)[/tex], [tex]\( P(B) = 0.3 \)[/tex], and [tex]\( P(A \text{ OR } B) = 0.95 \)[/tex], find [tex]\( P(A \text{ AND } B) \)[/tex].

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Answer :

GinnyAnswer
To find [tex]\( P(A \text{ AND } B) \)[/tex], you can use the formula for the probability of the union of two events. The formula states:

[tex]\[ P(A \text{ OR } B) = P(A) + P(B) - P(A \text{ AND } B) \][/tex]

Given:
- [tex]\( P(A) = 0.9 \)[/tex]
- [tex]\( P(B) = 0.3 \)[/tex]
- [tex]\( P(A \text{ OR } B) = 0.95 \)[/tex]

We need to find [tex]\( P(A \text{ AND } B) \)[/tex]. Let's denote it as [tex]\( P(A \cap B) \)[/tex].

Rearrange the formula to solve for [tex]\( P(A \cap B) \)[/tex]:

[tex]\[ P(A \cap B) = P(A) + P(B) - P(A \text{ OR } B) \][/tex]

Substitute the given probabilities into the formula:

[tex]\[ P(A \cap B) = 0.9 + 0.3 - 0.95 \][/tex]

Simplify the expression:

[tex]\[ P(A \cap B) = 1.2 - 0.95 \][/tex]

[tex]\[ P(A \cap B) = 0.25 \][/tex]

Therefore, the probability that both events [tex]\( A \)[/tex] and [tex]\( B \)[/tex] occur, [tex]\( P(A \text{ AND } B) \)[/tex], is 0.25.

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