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# Expected value

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I need help with these

Suppose we have a bag with 10 slips of paper in it. Eight slips have a 3 on them and the other two have a 5 on them.

(a) What is the expected value of the number shown when we draw a single slip of paper?
(b) What is the expected value of the number shown if we add one additional 3 to the bag?
(c) What is the expected value of the number shown if we add two additional 3's (instead of just one) to the bag?

Mar 5, 2023

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(a) The expected value of the number shown when we draw a single slip of paper is:

E(X) = (8/10) * 3 + (2/10) * 5 = 2.8

(b) If we add one additional 3 to the bag, there will be a total of 9 slips with a 3 on them and 2 slips with a 5 on them. The expected value of the number shown is:

E(X) = (9/11) * 3 + (2/11) * 5 = 2.81

(c) If we add two additional 3's to the bag, there will be a total of 10 slips with a 3 on them and 2 slips with a 5 on them. The expected value of the number shown is:

E(X) = (10/12) * 3 + (2/12) * 5 = 2.83

Note that the expected value increases as we add more slips with a 3 on them to the bag. This makes sense, since the mode of the distribution is 3, and adding more slips with a value of 3 makes it more likely that we will draw a 3.

Mar 5, 2023