If A and B are two independent events such th | vedclass

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(A) Since $A$ and $B$ are independent events,$P(A \cap B) = P(A)P(B)$.Given $P(A \cap \bar{B}) = P(A)P(\bar{B}) = P(A)(1 - P(B)) = \frac{3}{25}$.Given

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$\frac{12}{25}$

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(A) Since $A$ and $B$ are independent events,$P(A \cap B) = P(A)P(B)$.Given $P(A \cap \bar{B}) = P(A)P(\bar{B}) = P(A)(1 - P(B)) = \frac{3}{25}$.Given $P(\bar{A} \cap B) = P(\bar{A})P(B) = (1 - P(A))P(B) = \frac{8}{25}$.Let $P(A) = a$ and $P(B) = b$. Then $a(1 - b) = \frac{3}{25} \Rightarrow a - ab = \frac{3}{25}$ and $b(1 - a) = \frac{8}{25} \Rightarrow b - ab = \frac{8}{25}$.Subtracting the two equations: $(b - ab) - (a - ab) = \frac{8}{25} - \frac{3}{25} \Rightarrow b - a = \frac{5}{25} = \frac{1}{5}$.Thus,$b = a + \frac{1}{5}$.Substituting into $a - ab = \frac{3}{25}$: $a - a(a + \frac{1}{5}) = \frac{3}{25}$ $\Rightarrow a - a^2 - \frac{a}{5} = \frac{3}{25}$ $\Rightarrow \frac{4a}{5} - a^2 = \frac{3}{25}$.Multiplying by $25$: $20a - 25a^2 = 3 \Rightarrow 25a^2 - 20a + 3 = 0$.Factoring: $(5a - 3)(5a - 1) = 0$. So $a = \frac{3}{5}$ or $a = \frac{1}{5}$.Since $P(A) > 0.5$,we take $a = \frac{3}{5} = 0.6$.Then $b = \frac{3}{5} + \frac{1}{5} = \frac{4}{5} = 0.8$.Finally,$P(A \cap B) = ab = \frac{3}{5} 	imes \frac{4}{5} = \frac{12}{25}$.

If $A$ and $B$ are two independent events such that $P(A) > 0.5$,$P(B) > 0.5$,$P(A \cap \bar{B}) = \frac{3}{25}$,and $P(\bar{A} \cap B) = \frac{8}{25}$,then $P(A \cap B)$ is:

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