You have a box with 6 coins. 3 of the coins are weighted so that they comes up heads with probability 0.1. (Category 1) 1 is weighted so that it comes up heads with probability 0.3. (Category 2) 2 are weighted so that they come up heads with probability 0.7. (Category 3)

Questions

1. What is the probability of heads, if you pick a coin at random and flip it?
2. Suppose that you pick three coins at random from the box and you flip each of them 10 times. What is the expected total number of heads? Justify your answer. (Hint: Use random variables. This is easy, once you have done part (1). If you start enumerating the possible combinations of categories, you are on the wrong track.)
3. Suppose that you pick a coin at random and flip it and it comes up heads. What are the probabilities of each of the categories? What is the probability that it will come up heads again if you flip it again?
4. Suppose that you pick a coin at random, flip it twice, and it comes up heads both times. What are the probabilities of each of the categories? What is the probability that it will come up heads again if you flip it again?