Question

Suppose thatA,B, andCare three events such thatAandBare disjoint,AandCare independent, andBandCare independent. Suppose also that\({\bf{4Pr}}\left( {\bf{A}} \right){\bf{ = 2Pr}}\left( {\bf{B}} \right){\bf{ = Pr}}\left( {\bf{C}} \right)\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {{\bf{A}} \cup {\bf{B}} \cup {\bf{C}}} \right){\bf{ = 5Pr}}\left( {\bf{A}} \right)\). Determine thevalue of\({\bf{Pr}}\left( {\bf{A}} \right)\)

Short answer

The probability for event A is 1/6

Step-by-step solution

Given information

\(4\Pr \left( A \right) = 2\Pr \left( B \right) = \Pr \left( C \right)\,\,and\,\,\Pr \left( {A \cup B \cup C} \right) = 5\Pr \left( A \right)\)

Finding the probability of event A

\(\begin{aligned}{}\Pr \left( {A \cup B \cup C} \right) &= \Pr \left( A \right) + \Pr \left( B \right) + \Pr \left( C \right) - \Pr \left( {A \cap B} \right) - \Pr \left( {B \cap C} \right) - \Pr \left( {C \cap A} \right) + \Pr \left( {A \cap B \cap C} \right)\\5\Pr \left( A \right) &= \Pr \left( A \right) + 2\Pr \left( A \right) + 4\Pr \left( A \right) - 0 - \Pr \left( B \right)\Pr \left( C \right) - \Pr \left( A \right)\Pr \left( C \right) + 0\\5\Pr \left( A \right) &= 7\Pr \left( A \right) - \Pr \left( B \right)\Pr \left( C \right) - \Pr \left( A \right)\Pr \left( C \right)\\5\Pr \left( A \right) &= 7\Pr \left( A \right) - 2\Pr \left( A \right) \times 4\Pr \left( A \right) - \Pr \left( A \right) \times 4\Pr \left( A \right)\\5\Pr \left( A \right) &= 7\Pr \left( A \right) - 12{\left( {\Pr \left( A \right)} \right)^2}\end{aligned}\)

\(\begin{aligned}{}12{\left( {\Pr \left( A \right)} \right)^2} &= 2\Pr \left( A \right)\\6\Pr \left( A \right)& = 1\\\Pr \left( A \right) &= \frac{1}{6}\end{aligned}\)

The probability for event A is 1/6