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Conditional probability: the following problem is loosely based on the television game show Let's Make a Deal. At the end of the show, a contestant is asked to choose one of three large boxes, where one box contains a fabulous prize and the other two boxes contain lesser prizes. After the contestant chooses a box, Monty Hall, the host of the show, opens one of the two boxes containing smaller prizes. (In order to keep the conclusion suspenseful, Monty does not open the box selected by the contestant.) Monty offers the contestant the opportunity to switch from the rhosen box to the remaining unopened box. Should the contestant switch or stay with the original choice? Calculate the probability that the contestant wins under each strategy. This is an exercise in being clear about the information that should be conditioned on when constructing a probability judgment. See Selvin (1975) and Morgan et al. (1991) for further discussion of this problem.

Subjective probability: discuss the following statement. 'The probability of event \(\mathrm{E}\) is considered "subjective" if two rational persons A and B can assign unequal probabilities to \(\mathrm{E}, P_{A}(E)\) and \(P_{B}(E)\). These probabilities can also be interpreted as "conditional": \(P_{A}(E)=P\left(E \mid I_{A}\right)\) and \(P_{B}(E)=\) \(P\left(E \mid I_{B}\right)\), where \(I_{A}\) and \(I_{B}\) represent the knowledge available to persons A and B, respectively.' Apply this idea to the following examples. (a) The probability that a ' 6 ' appears when a fair die is rolled, where \(A\) observes the outcome of the die roll and B does not. (b) The probability that Brazil wins the next World Cup, where A is ignorant of soccer and \(\mathrm{B}\) is a knowledgeable sports fan.