25 we just used bayes formula perhaps without even

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Unformatted text preview: aw of total probability uses terms of the form P (A|Ei ). In many instances, as illustrated in the following examples, the values P (A|Ei ) are specified by the probability model, and (2.13) or (2.14) are used for calculating P (Ei |A). Example 2.10.1 There are three dice in a bag. One has one red face, another has two red faces, and the third has three red faces. One of the dice is drawn at random from the bag, each die having an equal chance of being drawn. The selected die is repeatedly rolled. (a) What is the probability that red shows on the first roll? (b) Given that red shows on the first roll, what is the conditional probability that red shows on the second roll? (c) Given that red shows on the first three rolls, what is the conditional probability that the selected die has red on three faces? 2.10. THE LAW OF TOTAL PROBABILITY, AND BAYES FORMULA 49 Solution: Let Ei be the event that the die with i red faces is drawn from the bag, for i =1, 2, or 3. Let Rj denote the event that red shows on the j th role of the die. (a) By the law of total pr...
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