1. Suppose Emi has 2 coins: one is a typical fair coin, the other is rigged and has "Heads" on both sides. Emi picks one at random (50% chance of picking each) and is curious to see which one she got. But there's a catch - Emi is not allowed to turn the coin over in her hand. She can only move the coin by tossing it in the air. After the coin lands, she can look to see what it shows.
a. Suppose Emi tosses the coin once and sees that the result is Heads. What are the odds that she has the rigged coin? What about if the result is "Tails"?
b. Now suppose Emi repeats this action 8 times, and each time the result is "Heads" case.
c. Consider again the question from part b, but this time Emi picked the coin from an urn of 1,000 coins, which contained 999 regular coins and 1 Heads/Heads coin. If she threw the coin 8 times and the result was Heads each time, what is the probability she has the rigged coin?
2. Suppose you have two 6-sided dice - one is fair, and the other is loaded, with probabilities Pr1 = Pr2 = Pr3 = Pr4 = 1/10, Pr5 = 1/5, Pr6 = 2/5.
a. Let X = the number given by the first die. Write down its probability table and calculate its mean and variance.
b. Do the same for variable Y, where Y = the number given by the second die.
c. Do the same for variable Z, which is defined as the sum of the two dice. Hint: to build the probability table for Z, you will need to first tabulate all the possible combinations from throwing the two dice.
3. Suppose you are playing Dungeons & Dragons with your friends and are supposed to throw a 10-sided die. If you get 9 or 10, your character will be able to collect a valuable artifact, which won't be available to the other players. However, you lost the 10-sided die. All you have are one 6-sided die and one 4-sided die. (You can assume all dice are fair unless mentioned otherwise)
a. Your friend Adam proposes the following: you throw the two dice and consider their sum as if it was the result of the 10-sided die (i.e. you "win" the toss if the sum is 9 or 10). Would you accept this solution?
b. Bob says that Adam's solution is not fair to you and suggests that the sum of the two dice should be compared to 8 instead (i.e. you win if the sum is 8 or greater). Would you accept this solution? Would the others?
c. Cora suggests that you throw the 6-sided die twice and count it as a win if the sum of the dice is at least 10. What odds does this give you?
d. Can you think of some other (better) solution?
