Question

I. Box I contains 3 red and 2 blue marbles while Box II contains 2 red and 8 blue marbles. A fair coin istossed. If the coin turns up heads, a marble is chosen from Box I; if it turns up tails, a marble is chosen from Box II. Let R denote the event “a red marble is chosen” while I and II denote the events that Box I and Box II are chosen, respectively. Find the probability that a red marble is chosen. 

Answer 1

Let $B_1$ denote the event "Box I is chosen", $B_2$ denote the event "Box II is chosen".

By the Law of total Probability

$$P(R)=P(R|B_1)P(B_1)+P(R|B_2)P(B_2)$$

$$P(R)=\dfrac{1}{2}(\dfrac{3}{5})+\dfrac{1}{2}(\dfrac{2}{10})=\dfrac{2}{5}$$

The probability that a red marble is chosen is $\dfrac{2}{5} .$