Here is a game. If you win the game, you get a dollar; else, you lose one. What is the probability of winning the game?
The game involves a fair coin and two urns.
Urn 1: 3 red balls; 1 blue ball.
Urn 2: 1 red ball; 3 blue balls.
You toss the coin first. If heads, you draw a ball from urn 1 and if tails, urn 2. Drawing a red ball wins the game.
The marginal probability of getting a head is 1/2, and getting a red ball from Urn 1 = 3/4. Therefore, the joint probability of getting a red ball from Urn 1 is (1/2)x(3/4) = (3/8). Similarly, the joint probability of getting a red ball from Urn 2 is (1/2)x(1/4) = (1/8). The overall probability of drawing a red is
(3/8) + (1/8) = (4/8) = (1/2), same as flipping a coin.
