Question

Two bags: A (2W, 3R), B (4W, 5R). One red drawn. Find probability it came from B.

Answer 1

Step 1: Given probabilities:
\[ P(B) = \frac{1}{2}, \quad P(A) = \frac{1}{2}, \quad P(R|A) = \frac{3}{5}, \quad P(R|B) = \frac{5}{9} \]
Step 2: Using Bayes' Theorem: \[ P(B|R) = \frac{P(B) \cdot P(R|B)}{P(A) \cdot P(R|A) + P(B) \cdot P(R|B)} \] Step 3: Substitute the values: \[ P(B|R) = \frac{\frac{1}{2} \cdot \frac{5}{9}}{\frac{1}{2} \cdot \frac{3}{5} + \frac{1}{2} \cdot \frac{5}{9}} \] Step 4: Simplify numerator and denominator:

• Numerator = \( \frac{5}{18} \)
• Denominator = \( \frac{3}{10} + \frac{5}{18} = \frac{27}{90} + \frac{25}{90} = \frac{52}{90} \)

\[ P(B|R) = \frac{5}{18} \div \frac{52}{90} = \frac{5}{18} \cdot \frac{90}{52} = \frac{450}{936} = \frac{25}{52} \]
Final Answer: \[ \boxed{\frac{25}{52}} \]