2. (16 points) James is upset that you beat him at Ultimate Mecha Strike III. Let's say he chooses a character that has two different movesets. These movesets are assigned randomly when he starts the game. In the first set, there are two punches and six kicks, while in the second set, there are four punches and four kicks. (a) (8 points) James uniformly and randomly chooses between the two movesets and then performs a random move. Given that the random move was a punch what is the probability that it came from moveset 1? Solution: (b) (8 points) Now imagine this character carries a special move that behaves like a standard six-sided die. If they roll a 3 or a 5, they use moveset 1; otherwise, they use moveset 2. Now, let's say they just performed a punch. What's the probability that this punch came from moveset 2 after rolling the die? Solution: 3. (12 points) Suppose you and your friend each choose at random an integer between 1 and 8, inclusive. For example, some possibilities are (3, 7), (7, 3), (4, 4), (8, 1), where your number is written first and your friend's number second. Find the following probabilities: (a) (3 points) p(you pick 3 and your friend picks 7) Solution: (b) (3 points) p(sum of the two numbers picked is