3. A certain video game can be played on easy mode or hard mode. On easy mode, every time you play, you have a 90% chance of winning. On hard mode, every time you play, you have only a 20% chance of winning. You sit down to play, but you quickly realize that your sibling has been messing with your gaming system. You don't know whether they set the game on easy mode or hard mode. You estimate a 2/3 probability that they set the game on hard mode and a 1/3 probability that they set the game on easy mode. Suppose that you play two games. Let Wj be the event that you win the first game and We be the event that you win the second game. Let E be the event that the game is on easy mode, and let / be the event that the game is on hard mode. (a) Find P(W1 ) and P(W2). (b) Are Wi and Wa conditionally independent given E? Are Wi and W2 con- ditionally independent given II? (c) Are Wi and We independent? If not, explain why knowledge of Wi affects the probability of Wa