If \(X\) and \(Y\) are two independent events such that \(P (X) = \frac{1}{8}\) and \(P (X ⋃ Y) = \frac{5}{8}\), find \(P (Y)\).
\(P(X⋃Y)=\frac{5}{8}\)
\(P(X⋂Y)=P(X)\times P(Y)\)
Since X and Y are independent events, the probability of their union (X ⋃ Y) can be calculated as:
\(P(X⋃Y)=P(X)+P(Y)-P(X⋂Y)\)
\(=\frac{5}{8}=\frac{1}{8}+P(Y)-\frac{1}{8}\times P(Y)\)
\(=\frac{5}{8}-\frac{1}{8}=P(Y)-\frac{1}{8}\times P(Y)\)
\(=\frac{1}{2}=P(Y)(1-\frac{1}{8})\)
\(=\frac{1}{2}=P(Y)(\frac{7}{8})\)
\(=P(Y)=\frac{1}{2}÷\frac{7}{8}\)
\(∴P(Y)=\frac{1}{2}x\frac{8}{7}=\frac{4}{7}\)
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