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Chapter 2: Q5SE (page 90)
Suppose that in 10 rolls of a balanced die, the number 6 appeared exactly three times. What is the probability that the first three rolls each yielded the number 6?
The probability that the first three rolls each yields the number 6 is 0.001292
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For any two events A and B with Pr(B) > 0, prove that Pr(Ac|B) = 1 鈭� Pr(A|B).
Use the prior probabilities in Example 2.3.8 for the events\({B_1},...,{{\bf{B}}_{11}}\). Let \({E_1}\)be the event that the first patient is a success. Compute the probability of \({E_1}\) and explain why it is so much less than the value computed in Example 2.3.7.
Consider again the conditions of Example 2.3.4 in this section, in which an item was selected at random from a batch of manufactured items and was found to be defective.For which values ofi(i=1,2,3) is the posterior probability that the item was produced by machineMilarger than the prior probability that the item was producedby machineMi?
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 box contains five coins and that for each coin there is a different probability that a head will be obtained when the coin is tossed. Let \({{\bf{p}}_{\bf{i}}}\)denote the probability of a head when theith coin is tossed \({\bf{i = }}\left( {{\bf{1, \ldots ,5}}} \right)\) and suppose that \({{\bf{p}}_{\bf{1}}}{\bf{ = 0}}\),\({{\bf{p}}_{\bf{2}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}\) ,\({{\bf{p}}_{\bf{3}}}{\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}\) ,\({{\bf{p}}_{\bf{4}}}{\bf{ = }}\frac{{\bf{3}}}{{\bf{4}}}\) , and \({{\bf{p}}_{\bf{5}}}{\bf{ = 1}}\).
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