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A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross
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There are 2 coins in a bin. When one of them is flipped,

Chapter 4, Problem 6STE

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QUESTION:

Problem 6STE

There are 2 coins in a bin. When one of them is flipped, it lands on heads with probability .6, and when the other is flipped, it lands on heads with probability .3. One of these corns is to be randomly chosen and then flipped. Without knowing which coin is chosen, you can bet any amount up to $10, and you then either win that amount if the coin comes up heads or lose it if it comes up tails. Suppose, however, that an insider is willing to sell you, for an amount C, the information as to which coin was selected. What is your expected payoff if you buy this information? Note that if you buy it and then bet x, you will end up either winning x − C or −x − C(that is, losing x + C in the latter case). Also, for what values of C does it pay to purchase the information?

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QUESTION:

Problem 6STE

There are 2 coins in a bin. When one of them is flipped, it lands on heads with probability .6, and when the other is flipped, it lands on heads with probability .3. One of these corns is to be randomly chosen and then flipped. Without knowing which coin is chosen, you can bet any amount up to $10, and you then either win that amount if the coin comes up heads or lose it if it comes up tails. Suppose, however, that an insider is willing to sell you, for an amount C, the information as to which coin was selected. What is your expected payoff if you buy this information? Note that if you buy it and then bet x, you will end up either winning x − C or −x − C(that is, losing x + C in the latter case). Also, for what values of C does it pay to purchase the information?

ANSWER:

Solution 6STE

Step1 of 2:

From the given problem we have two fair coins in a bin. Suppose one of them is flipped, it lands on heads with probability 0.6, and when the other is flipped, it lands on heads with probability 0.3. The betting amount is $10, here we win when the coin comes up with heads, and we lose when coin come up with tails.

We need to find expected payoff, when we buy this information.

Step2 of 2:

Let us consider a discrete random variable ‘Y’, and ‘x’ is betting that wins with probability

(1 - p). Then, Expected winning is:

                                                                                 ..….(1)   

 

Consider,

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