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Chapter 4: Q.4.14 (page 174)
On average, hurricanes hit a certain region in a year. What is the probability that there will be or fewer hurricanes hitting this year?
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A communications channel transmits the digits and However, due to static, the digit transmitted is incorrectly received with probability Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit instead of and 11111 instead of If the receiver of the message uses 鈥渕ajority鈥� decoding, what is the probability that the message will be wrong when decoded? What independence assumptions are you making?
There are types of coupons. Independently of the types of previously collected coupons, each new coupon collected is of typewith probability , . If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of coupons.)
Two coins are to be 铿俰pped. The 铿乺st coin will land on heads with probability ., the second with probability .. Assume that the results of the 铿俰ps are independent, and let X equal the total number of heads that result. (a) Find P{X =}. (b) Determine E[X].
In the game of Two-Finger Morra, players show or fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.
(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the possibilities is equally likely, what are the possible values of and what are their associated probabilities?
(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up or fingers, what are the possible values of and their associated probabilities?
Suppose that a biased coin that lands on heads with probability is flipped times. Given that a total of heads results, find the conditional probability that the first outcomes are
(a) (meaning that the first flip results in heads, the second is tails, and the third in tails);
(b)
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