Chapter 2: Q14E (page 86)
Consider the conditions of Exercise 23 in Sec. 2.2 again. Assume that Pr(B)=0.4. LetAbe the event that exactly 8 out of 11 programs compiled. Compute the conditional probability ofBgivenA.
The conditional probability of Bgiven Ais 0.3874
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Suppose that three red balls and three white balls are thrown at random into three boxes and and that all throws are independent. What is the probability that each box contains one red ball and one white ball?
Suppose that there is a probability of\(\frac{1}{{50}}\)that you will win a certain game. If you play the game 50 times, independently, what is the probability that you will win at least once?
Three prisonersA,B, andCon death row know that exactly two of them are going to be executed, but they do not know which two. PrisonerAknows that the jailer will not tell him whether or not he is going to be executed. He, therefore, asks the jailer to tell him the name of one Prisoner other thanAhimself who will be executed. The jailer responds thatBwill be executed. Upon receiving this response, PrisonerAreasons as follows: Before he spoke to the jailer, the probability was \(\frac{2}{3}\) that he would be one of the two prisoners executed. After speaking to the jailer, he knows that either he or PrisonerCwill be the other one to be executed. Hence, the probability that he will be executed is now only \(\frac{1}{2}\). Thus, merely by asking the jailer his question, the Prisoner reduced the probability that he would be executed \(\frac{1}{2}\)because he could go through exactly this same reasoning regardless of which answer the jailer gave. Discuss what is wrong with prisonerA’s reasoning.
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 \({{\bf{p}}_{\bf{5}}}{\bf{ = 1}}\).
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.
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