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A and B are events such that P (A) = 0. ...

A and B are events such that P (A) = 0. 4 , P (B) =0. 3 and ` P (A cup B) = 0 . 5` . Then ` P (B cap A)` equals

A

`(2)/(3)`

B

`(1)/(2)`

C

`(3)/(10)`

D

`(1)/(5)`

Text Solution

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The correct Answer is:
To find \( P(A \cap B) \) (the probability of both events A and B occurring), we can use the formula for the probability of the union of two events: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Given: - \( P(A) = 0.4 \) - \( P(B) = 0.3 \) - \( P(A \cup B) = 0.5 \) We can substitute the known values into the formula: \[ 0.5 = 0.4 + 0.3 - P(A \cap B) \] Now, let's simplify the equation: 1. First, add \( P(A) \) and \( P(B) \): \[ 0.4 + 0.3 = 0.7 \] 2. Substitute this sum back into the equation: \[ 0.5 = 0.7 - P(A \cap B) \] 3. Rearranging the equation to isolate \( P(A \cap B) \): \[ P(A \cap B) = 0.7 - 0.5 \] 4. Calculate the right-hand side: \[ P(A \cap B) = 0.2 \] Thus, the probability of both events A and B occurring is: \[ P(A \cap B) = 0.2 \]
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