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Question
In a competition A, B and C are participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.
Solution
\[P\left( \text{ A wins } \right) = P\left( A \right)\]
\[P\left( \text{ B wins } \right) = P\left( B \right) = \frac{P\left( A \right)}{2}\]
\[P\left( \text{ C wins } \right) = P\left( C \right) = \frac{P\left( B \right)}{2} = \frac{P\left( A \right)}{4}\]
\[\text{ Now } , \]
\[P\left( A \right) + P\left( B \right) + P\left( C \right) = 1\]
\[ \Rightarrow P\left( A \right) + \frac{P\left( A \right)}{2} + \frac{P\left( A \right)}{4} = 1\]
\[ \Rightarrow P\left( A \right)\left[ \frac{4 + 2 + 1}{4} \right] = 1\]
\[ \Rightarrow P\left( A \right) = \frac{4}{7}\]
\[P\left( \text{ A loses } \right) = P\left( \bar{A} \right)\]
\[ = 1 - \frac{4}{7}\]
\[ = \frac{3}{7}\]
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