If A and B are two events such that P A- B -5-6- P A- B -1-3-P B -1-2- then the events A and B are

The correct option is B independentGiven P(A∪ B)=56,P(A∩ B)=13,P(B̅)= 12 P ( B )=1 P ( B )=1 1/2=1/2 #160; #160; P ( A∪ B )=P ( A )+P ( B ...

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Byju's Answer

Standard X

Mathematics

Addition Theorem of Probability for 2 Events

If A and ...

Question

If $A$ and $B$ are two events such that $P (A \cup B) = \frac{5}{6},$ $P (A \cap B) = \frac{1}{3}, P (¯ ¯¯ ¯ B) = \frac{1}{2},$ then the events $A$ and $B$ are

dependent

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independent

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mutually exclusive

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none of these.

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Solution

The correct option is B independent
Given $P (A \cup B) = \frac{5}{6}, P (A \cap B) = \frac{1}{3}, P (¯ B) = \frac{1}{2}$

$P (B) = 1 - P (¯ ¯¯ ¯ B) = 1 - \frac{1}{2} = \frac{1}{2}$

$P (A \cup B) = P (A) + P (B) - P (A \cap B)$

$\frac{5}{6} = P (A) + \frac{1}{2} - \frac{1}{3}$

$P (A) = \frac{2}{3} .$

$∴ P (A) P (B) = \frac{2}{3} \times \frac{1}{2} = \frac{1}{3} = P (A \cap B) .$

Hence $A$ and $B$ are independent.

Suggest Corrections

Similar questions

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If A and B are two events such that P(A∪B)=56, P(A∩B)=13,P(¯¯¯¯B)=12, then the events A and B are

The correct option is B independentGiven P(A∪B)=56,P(A∩B)=13,P(¯B)=12P(B)=1−P(¯¯¯¯B)=1−12=12 P(A∪B)=P(A)+P(B)−P(A∩B) 56=P(A)+12−13 P(A)=23. ∴P(A)P(B)=23×12=13=P(A∩B). Hence A and B are independent.

If $A$ and $B$ are two events such that $P (A \cup B) = \frac{5}{6},$ $P (A \cap B) = \frac{1}{3}, P (¯ ¯¯ ¯ B) = \frac{1}{2},$ then the events $A$ and $B$ are