Problem 10 You have a box with 4 coins, all weighted. It is given that: . 1 coin comes up heads with probability 0.1 (Category 1). . 1 coin comes up heads with probability 0.5 (Category 2). . 2 coins come up heads with probability 0.8 (Category 3). 1. You pick a coin out of the box at random and flip it. What is the probability that it will come up heads? 2. You pick a coin out of the box and flip it twice. What is the probability of two heads? 3. You pick two coins out of the box together (at the same time) and flip each of them once. What is the probability of two heads? 4. You pick a coin of the box at random, flip it, put it back, again pick a coin at random, and flip it. What is the probability of two heads? 5 . You pick a coin out of the box at random and flip it. It comes up heads. What is the probability that it is Category 1? Category 2? 6. You pick a coin out of the box at random and flip it. It comes up tails. What is the probability that it is Category 1? Category 2? 7 . You pick a coin out of the box at random and flip it. It comes up heads. You flip it again. What is the probability that the second flip will be heads? 8. You pick a coin out of the box at random and flip it twice. It comes up heads both times. What is the probability that it is Category 1? Category 2? 9. You pick a coin out of the box at random and flip it 10 times. What is the expected number of heads? (Hint: This is easy, once you've worked out part A. If you start calculating the probability distribution over the number of heads, you're on the wrong track.)