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If A, B, C are three events such that P(...

If A, B, C are three events such that `P(B)=(4)/(5), P(A nn B nn C^(c ))=(1)/(4) and P(A^(c )nnBnnC^(c ))=(1)/(3)`, then `P(BnnC)` is equal to

A

`(11)/(60)`

B

`(1)/(5)`

C

`(13)/(60)`

D

`(1)/(4)`

Text Solution

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The correct Answer is:
To solve the problem, we need to find the probability of the intersection of events B and C, denoted as \( P(B \cap C) \). We are given the following information: 1. \( P(B) = \frac{4}{5} \) 2. \( P(A \cap B \cap C^c) = \frac{1}{4} \) 3. \( P(A^c \cap B \cap C^c) = \frac{1}{3} \) ### Step-by-Step Solution: **Step 1: Understand the Events and Their Relationships** We need to analyze the relationships between the events A, B, and C. We can use a Venn diagram to visualize the intersections of these events. **Step 2: Set Up the Equation for \( P(B) \)** The probability of event B can be expressed in terms of the intersections with A and C: \[ P(B) = P(B \cap C) + P(A \cap B \cap C^c) + P(A^c \cap B \cap C^c) \] **Step 3: Substitute the Known Values** Now, substituting the known values into the equation: \[ P(B) = P(B \cap C) + P(A \cap B \cap C^c) + P(A^c \cap B \cap C^c) \] \[ \frac{4}{5} = P(B \cap C) + \frac{1}{4} + \frac{1}{3} \] **Step 4: Find a Common Denominator** To simplify the equation, we need to find a common denominator for the fractions \( \frac{1}{4} \) and \( \frac{1}{3} \). The least common multiple of 4 and 3 is 12. Convert the fractions: \[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{3} = \frac{4}{12} \] **Step 5: Substitute Back into the Equation** Now substituting back into the equation: \[ \frac{4}{5} = P(B \cap C) + \frac{3}{12} + \frac{4}{12} \] \[ \frac{4}{5} = P(B \cap C) + \frac{7}{12} \] **Step 6: Isolate \( P(B \cap C) \)** Now, isolate \( P(B \cap C) \): \[ P(B \cap C) = \frac{4}{5} - \frac{7}{12} \] **Step 7: Find a Common Denominator for the Final Calculation** The common denominator for 5 and 12 is 60. Convert the fractions: \[ \frac{4}{5} = \frac{48}{60}, \quad \frac{7}{12} = \frac{35}{60} \] **Step 8: Calculate \( P(B \cap C) \)** Now, perform the subtraction: \[ P(B \cap C) = \frac{48}{60} - \frac{35}{60} = \frac{13}{60} \] ### Final Answer: Thus, the probability \( P(B \cap C) \) is \( \frac{13}{60} \).
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