UPSC NDA I 2018 Mathematics Solved Paper

Concept:The probability of drawing a white ball is found by considering both bags, since a bag is chosen at random first.Explanation:Step 1: Define the events.Let $E_1$ be the event "choosing bag one" and $E_2$ be the event "choosing bag two".Let $A$ be the event "drawing a white ball".Step 2: Assign probabilities for choosing a bag.Since a bag is chosen at random, $P(E_1) = \frac{1}{2}$ and $P(E_2) = \frac{1}{2}$.Step 3: Find the probability of drawing a white ball from each bag.Bag one has 3 white and 2 black balls (total 5). So $P(A|E_1) = \frac{3}{5}$.Bag two has 5 white and 3 black balls (total 8). So $P(A|E_2) = \frac{5}{8}$.Step 4: Apply the law of total probability.$P(A) = P(E_1) \cdot P(A|E_1) + P(E_2) \cdot P(A|E_2)$$P(A) = \left(\frac{1}{2} \times \frac{3}{5}\right) + \left(\frac{1}{2} \times \frac{5}{8}\right)$$P(A) = \frac{3}{10} + \frac{5}{16}$Step 5: Add the fractions.Find a common denominator, 80.$\frac{3}{10} = \frac{24}{80}$, $\frac{5}{16} = \frac{25}{80}$.$P(A) = \frac{24 + 25}{80} = \frac{49}{80}$.Answer:$\frac{49}{80}$ (Option B).