For each of the following, describe in detail (i.e., write out the algorithm) a simulation that would yield an approximate answer to the given question. Assume that the “equipment” available to simulate is an unlimited supply of coins and playing cards.
Suppose you flip a fair coin four times, recording each flip in order. Which outcome is more common, or ?
What is the average number of flips needed to flip a fair coin until two successive heads are observed?
An unfair 6-sided die has sides marked with the numbers 1, 2, 3, 4, 5 and 6. The probability of rolling a 1 is while the probability of rolling each of the other numbers is If the die is rolled twice, what is the probability that the sum of the two rolls is 7?
As we age, the probability that we survive one more year changes. For example, the probability of a 20-year-old woman dying is approximately 0.0004, while it is approximately 0.003 for a 50-year-old woman. Suppose that the probability of someone dying remains fixed from birth, and that this probability is 0.0004. What is the average lifespan?
A problem posed by the American statistician Steve Selvin can be stated as follows: Suppose that you are on a game show in which the emcee presents three doors to you. Behind one door is a brand new car, while the other two each hide a goat. The emcee asks you to pick a door, and let’s you know that you will get to keep what is behind the door your choose. After you pick a door, the emcee opens one of the two doors you did not pick to reveal a goat. The emcee then offers you opportunity to change your choice to the other closed door. Should you?