Let \(A\) and \(B\) be independent events such that \(P(A) = p, P(B) = 2p\). The largest value of \(p\), for which \(P(\text{exactly one of } A, B \text{ occurs}) = \frac{5{9}\), is :}
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The expression for "exactly one happens" is symmetric. Always verify that your final \(p\) values allow for \(2p \leq 1\), as probabilities cannot exceed 1. Here \(2(5/12) = 5/6<1\), so it is valid.
Step 1: Understanding the Concept:
For exactly one of two independent events to occur, we sum the probabilities of \(A\) occurring without \(B\) and \(B\) occurring without \(A\). Since they are independent, \(P(A \cap B) = P(A)P(B)\). Step 2: Key Formula or Approach:
1. \(P(\text{Exactly one}) = P(A) + P(B) - 2P(A \cap B)\).
2. For independent events, \(P(A \cap B) = P(A)P(B)\). Step 3: Detailed Explanation:
Given \(P(A) = p\) and \(P(B) = 2p\).
\[ P(\text{Exactly one}) = p + 2p - 2(p \cdot 2p) = 3p - 4p^2 \]
We are given this probability is \(5/9\):
\[ 3p - 4p^2 = \frac{5}{9} \implies 27p - 36p^2 = 5 \]
\[ 36p^2 - 27p + 5 = 0 \]
Factoring the quadratic equation:
\[ 36p^2 - 12p - 15p + 5 = 0 \implies 12p(3p - 1) - 5(3p - 1) = 0 \]
\[ (12p - 5)(3p - 1) = 0 \implies p = \frac{5}{12} \text{ or } p = \frac{1}{3} \]
Comparing the values: \(5/12 \approx 0.416\) and \(1/3 \approx 0.333\).
The largest value is \(5/12\). Step 4: Final Answer:
The largest value of \(p\) is \(\frac{5}{12}\).
