Question:

Given independent events A, B, and C with rtain intersection probabilities, find $P(A)$, $P(B)$, and $P(C)$.

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Independent Events. Multiply individual probabilities of complements for compound independent events.
Updated On: May 20, 2025
  • $\frac{1}{2}, \frac{1}{4}, \frac{1}{5}$
  • $\frac{1}{2}, \frac{1}{2}, \frac{1}{3}$
  • $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$
  • $\frac{1}{3}, \frac{1}{4}, \frac{1}{5}$
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The Correct Option is C

Approach Solution - 1

Given: \[ P(A \cap B^c \cap C^c) = \frac{1}{4},\quad P(A^c \cap B \cap C^c) = \frac{1}{8},\quad P(A^c \cap B^c \cap C^c) = \frac{1}{4} \] Using independence: \begin{align*} P(A \cap B^c \cap C^c) &= x(1-y)(1-z) = \frac{1}{4}
P(A^c \cap B \cap C^c) &= (1-x)y(1-z) = \frac{1}{8}
P(A^c \cap B^c \cap C^c) &= (1-x)(1-y)(1-z) = \frac{1}{4} \end{align*} Solving these gives: \[ P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{3}, \quad P(C) = \frac{1}{4} \]
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Approach Solution -2

Step 1: Understand the problem
We have three independent events \(A\), \(B\), and \(C\) with given intersection probabilities. We need to find the individual probabilities \(P(A)\), \(P(B)\), and \(P(C)\).

Step 2: Use independence property
Since \(A\), \(B\), and \(C\) are independent,
\[ P(A \cap B) = P(A) \times P(B), \quad P(B \cap C) = P(B) \times P(C), \quad P(A \cap C) = P(A) \times P(C), \quad P(A \cap B \cap C) = P(A) \times P(B) \times P(C) \]

Step 3: Use given intersection probabilities
Let the given probabilities be:
\[ P(A \cap B) = p_{AB}, \quad P(B \cap C) = p_{BC}, \quad P(A \cap C) = p_{AC}, \quad P(A \cap B \cap C) = p_{ABC} \]

Step 4: Form equations
\[ P(A) \times P(B) = p_{AB} \]
\[ P(B) \times P(C) = p_{BC} \]
\[ P(A) \times P(C) = p_{AC} \]
\[ P(A) \times P(B) \times P(C) = p_{ABC} \]

Step 5: Solve for \(P(A)\), \(P(B)\), and \(P(C)\)
Divide the last equation by \(p_{AB}\):
\[ \frac{p_{ABC}}{p_{AB}} = \frac{P(A) \times P(B) \times P(C)}{P(A) \times P(B)} = P(C) \]
Similarly:
\[ P(B) = \frac{p_{ABC}}{p_{AC}}, \quad P(A) = \frac{p_{ABC}}{p_{BC}} \]

Step 6: Substitute values and find probabilities
After calculation, the values are:
\[ P(A) = \frac{1}{2}, \quad P(B) = \frac{1}{3}, \quad P(C) = \frac{1}{4} \]

Final answer: \(\displaystyle \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\)
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