Question

Let A and B be independent events with P (A) = 1/4 and P (A U B) = 2P (B) − P (A).
Find (a). P (B); (b).P (A|B); and (c). P (Bc|A).

Answer 1

Answer on Question #45523-Engineering-SolidWorks-CosmoWorks-Ansys

Let $A$ and $B$ be independent events with $P(A) = \frac{1}{4}$ and $P(A \cup B) = 2P(B) - P(A)$.

Find (a). P (B); (b). P (A | B); and (c). P (Bc | A).

Solution

(a). The probability of the union of $A$ and $B$

$$P (A \cup B) = P (A) + P (B) - P (A \cap B).$$

Since $A$ and $B$ are independent:

$$P (A \cap B) = P (A) \cdot P (B).$$

And we have:

$$\begin{array}{l} P (A \cup B) = P (A) + P (B) - P (A) \cdot P (B) = 2 P (B) - P (A). \\ \frac {1}{4} + P (B) - \frac {1}{4} \cdot P (B) = 2 P (B) - \frac {1}{4}. \\ P (B) = \frac {2}{5}. \end{array}$$

(b).

$$P (A | B) = \frac {P (A \cap B)}{P (B)} = \frac {P (A) \cdot P (B)}{P (B)} = P (A) = \frac {1}{4}.$$

(c). Since $A$ and $B$ are independent $A$ and $B^c$ are also independent. That's why

$$P (B ^ {c} | A) = \frac {P (B ^ {c} \cap A)}{P (A)} = \frac {P (B ^ {c}) \cdot P (A)}{P (A)} = P (B ^ {c}) = 1 - P (B) = 1 - \frac {2}{5} = \frac {3}{5}.$$

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