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Chapter 2: Q20SE (page 91)
Suppose that two players A and B take turns rolling a pair of balanced dice and that the winner is the first player who obtains the sum of 7 on a given roll of the two dice. If A rolls first, what is the probability that B will win?
The probability that B will win is 0.454545
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Consider again the three different conditions (a), (b), and (c) given in Exercise 2, but suppose now that p < 1/2. For which of these three conditions is there the greatest probability that gambler A will win the initial fortune of gambler B before he loses his own initial fortune?
Suppose thatkevents\({{\bf{B}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{B}}_{\bf{k}}}\)form a partition of the sample spaceS. For\({\bf{i = 1 \ldots k}}\), let\(\Pr \left( {{{\bf{B}}_{\bf{i}}}} \right)\)denote the prior probability of\({{\bf{B}}_{\bf{i}}}\). Also, for each eventAsuch that\({\bf{Pr}}\left( {\bf{A}} \right){\bf{ > 0}}\), let\({\bf{Pr}}\left( {{{\bf{B}}_{\bf{i}}}{\bf{|A}}} \right)\)denote the posterior probability of\({{\bf{B}}_{\bf{i}}}\)given that the eventAhas occurred. Prove that if\({\bf{Pr}}\left( {{{\bf{B}}_{\bf{1}}}{\bf{|A}}} \right){\bf{ < Pr}}\left( {{{\bf{B}}_{\bf{1}}}} \right)\)then\({\bf{Pr}}\left( {{{\bf{B}}_{\bf{i}}}{\bf{|A}}} \right){\bf{ > Pr}}\left( {{{\bf{B}}_{\bf{i}}}} \right)\)for at least one value of i\(\left( {{\bf{i = 2, \ldots ,k}}} \right)\).
Suppose that A and B are independent events such that\({\bf{Pr}}\left( {\bf{A}} \right){\bf{ = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{3}}}}\right.}\!\lower0.7ex\hbox{\({\bf{3}}\)}}\)and\({\bf{Pr}}\left( {\bf{B}} \right){\bf{ > 0}}\). What is the value of\({\bf{Pr}}\left( {{\bf{A}} \cup {{\bf{B}}{\bf{c}}}{\bf{|B}}} \right)\)?
Bus tickets in a certain city contain four numbers, U, V , W, and X. Each of these numbers is equally likely to be any of the 10 digits , and the four numbers are chosen independently. A bus rider is said to be lucky if\({\bf{U + V = W + X}}\). What proportion of the riders are lucky?
Consider a machine that produces items in sequence. Under normal operating conditions, the items are independent with a probability of 0.01 of being defective. However, it is possible for the machine to develop a 鈥渕emory鈥� in the following sense: After each defective item, and independent of anything that happened earlier, the probability that the next item is defective is\(\frac{{\bf{2}}}{{\bf{5}}}\). After each non-defective item, and independent of anything that happened earlier, the probability that the next item is defective is\(\frac{{\bf{1}}}{{{\bf{165}}}}\).
Please assume that the machine is operating normally for the whole time we observe or has a memory for the whole time we observe. LetBbe the event that the machine is operating normally, and assume that\({\bf{Pr}}\left( {\bf{B}} \right){\bf{ = }}\frac{{\bf{2}}}{{\bf{3}}}\). Let\({{\bf{D}}_{\bf{i}}}\)be the event that theith item inspected is defective. Assume that\({{\bf{D}}_{\bf{1}}}\)is independent ofB.
a. Prove that\({\bf{Pr}}\left( {{{\bf{D}}_{\bf{i}}}} \right){\bf{ = 0}}{\bf{.01}}\) for alli. Hint:Use induction.
b. Assume that we observe the first six items and the event that occurs is\({\bf{E = D}}_{\bf{1}}^{\bf{c}} \cap {\bf{D}}_{\bf{2}}^{\bf{c}} \cap {{\bf{D}}_3} \cap {{\bf{D}}_4} \cap {\bf{D}}_{\bf{5}}^{\bf{c}} \cap {\bf{D}}_{\bf{6}}^{\bf{c}}\). The third and fourth items are defective, but the other four are not. Compute\({\bf{Pr}}\left( {{\bf{B}}\left| {\bf{E}} \right.} \right)\).
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