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Step-by-step Solution
Step 1: Understand the Problem
We have two independent events A and B with probabilities $P(A) = p$ and $P(B) = 2p$. We are also given that the probability of exactly one of A or B occurring (but not both) is $ \frac{5}{9} $. We need to find the largest possible value of $p$ that satisfies this condition.
Step 2: Express the Probability of Exactly One Event Occurring
The probability of exactly one of A or B occurring, denoted $P(\text{exactly one of A or B})$, is the sum of:
$P(A \cap \overline{B})$: A occurs and B does not occur.
$P(\overline{A} \cap B)$: B occurs and A does not occur.
Because A and B are independent, $P(A \cap \overline{B}) = P(A)\,P(\overline{B})$ and $P(\overline{A} \cap B) = P(\overline{A})\,P(B)$.
Hence,
$P(\text{exactly one of A or B})
= P(A)P(\overline{B}) + P(\overline{A})P(B).$
Step 3: Plug in the Given Values and Condition
Substitute $P(A)=p$ and $P(B)=2p$ into the formula, and use $P(\overline{A}) = 1 - p$ and $P(\overline{B}) = 1 - 2p$:
$p(1 - 2p) + (1 - p)(2p) = \frac{5}{9}.$
Step 4: Simplify and Form a Quadratic Equation
Distribute and combine like terms:
$p(1 - 2p) + (1 - p)(2p)
= p - 2p^2 + 2p - 2p^2
= p + 2p - 2p^2 - 2p^2
= 3p - 4p^2.$
The equation becomes
$3p - 4p^2 = \frac{5}{9}.$
Multiply through by 9 to clear the denominator and rearrange to form a standard quadratic:
$9(3p - 4p^2) = 5
\quad\Longrightarrow\quad
27p - 36p^2 = 5
\quad\Longrightarrow\quad
-36p^2 + 27p - 5 = 0.$
Or equivalently,
$36p^2 - 27p + 5 = 0.
Step 5: Solve the Quadratic Equation
Use the quadratic formula $p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=36$, $b=-27$, and $c=5$.
After solving, we get the roots
$p = \frac{1}{3} \quad \text{or} \quad p = \frac{5}{12}.$
Step 6: Identify the Largest Valid Value
The problem asks for the largest value of $p$. Between $\frac{1}{3}$ and $\frac{5}{12}$, the larger is
$\frac{5}{12}$.
Hence, the largest value of $p$ satisfying the given condition is
$\boxed{\frac{5}{12}}$.
