(A) Since events $A$ and $B$ are independent,we have $P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} p$. Using the addition theorem of probability: $P(A

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(A) Since events $A$ and $B$ are independent,we have $P(A \cap B) = P(A) \cdot P(B) = \frac{1}{2} p$. Using the addition theorem of probability: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ Substituting the given values: $\frac{3}{5} = \frac{1}{2} + p - \frac{1}{2} p$ $\frac{3}{5} = \frac{1}{2} + \frac{p}{2}$ Subtracting $\frac{1}{2}$ from both sides: $\frac{p}{2} = \frac{3}{5} - \frac{1}{2} = \frac{6-5}{10} = \frac{1}{10}$ Multiplying by $2$: $p = \frac{2}{10} = \frac{1}{5}$

Class 11 Mathematics Probability Odds in favor and odds against, Addition theorem on probability

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Given that the events $A$ and $B$ are such that $P(A) = \frac{1}{2}$,$P(A \cup B) = \frac{3}{5}$,and $P(B) = p$. Find $p$ if the events $A$ and $B$ are independent.

A
$\frac{1}{5}$
B
$\frac{1}{10}$
C
$\frac{2}{5}$
D
$\frac{3}{10}$

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Odds in favor and odds against, Addition theorem on probability

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Given that the events $A$ and $B$ are such that $P(A) = \frac{1}{2}$,$P(A \cup B) = \frac{3}{5}$,and $P(B) = p$. Find $p$ if the events $A$ and $B$ are independent.

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