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Chapter 4: Q. 4.40 (page 166)
On a multiple-choice exam with possible answers for each of the questions, what is the probability that a student will get or more correct answers just by guessing?
The probability that a student will get or more correct answers are
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A man claims to have extrasensory perception. As a test, a fair coin is flipped times and the man is asked to predict the outcome in advance. He gets out of correct. What is the probability that he would have done at least this well if he did not have ESP?
At time a coin that comes up heads with probability p is flipped and falls to the ground. Suppose it lands on heads. At times chosen according to a Poisson process with rate , the coin is picked up and flipped. (Between these times, the coin remains on the ground.) What is the probability that the coin is on its head side at time? Hint: What would be the conditional probability if there were no additional flips by time , and what would it be if there were additional flips by time ?
Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability , then he or she will receive a score of
We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability , what value of should he or she assert so as to maximize the expected score?
Suppose that the random variable is equal to the number of hits obtained by a certain baseball player in his next at-bats. If and, find
In response to an attack of missiles, antiballistic missiles are launched. The missile targets of the antiballistic missiles are independent, and each antiballstic missile is equally likely to go towards any of the target missiles. If each antiballistic missile independently hits its target with probability . use the Poisson paradigm to approximate the probability that all missiles are hit.
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