Chapter 7: Q1E (page 492)
Question: What is the expected number of heads that come up when a fair coin is flipped five times?
Answer:
The expected number of heads that come up when a fair coin is flipped five times is 2.5
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Question: A pair of dice is loaded. The probability that a appears on the first die is , and the probability that a appears on the second die is. Other outcomes for each die appear with probability . What is the probability of appearing as the sum of the numbers when the two dice are rolled?
Question: Suppose that we have prior information concerning whether a random incoming message is spam. In particular, suppose that over a time period, we find that s spam messages arrive and h messages arrive that are not spam.
a) Use this information to estimate p(S), the probability that an incoming message is spam, and\(p(\bar S)\), the probability an incoming message is not spam.
b) Use Bayes’ theorem and part (a) to estimate the probability that an incoming message containing the word w is spam, where p(w)is the probability that w occurs in a spam message and q(w) is the probability that w occurs in a message that is not spam.
Question: What is the probability that a randomly selected day of a leap year
(with 366 possible days) is in April?
Question: What is the expected sum of the numbers that appear when three fair dice are rolled?
Question: Devise a Monte Carlo algorithm that determines whether a permutation of the integers 1 through n has already been sorted (that is, it is in increasing order), or instead, is a random permutation. A step of the algorithm should answer “true” if it determines the list is not sorted and “unknown” otherwise. After k steps, the algorithm decides that the integers are sorted if the answer is “unknown” in each step. Show that as the number of steps increases, the probability that the algorithm produces an incorrect answer is extremely small. [Hint: For each step, test whether certain elements are in the correct order. Make sure these tests are independent.]
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