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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 7, 2023
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(a) The expected value of the number shown when we draw a single slip of paper is the weighted average of the possible outcomes, where the weights are the probabilities of each outcome. There are 8 slips with a 3 on them and 2 slips with a 5 on them, so the probability of drawing a 3 is 8/10 = 4/5 and the probability of drawing a 5 is 2/10 = 1/5. Therefore, the expected value is:

E(X) = (4/5) * 3 + (1/5) * 5 = 12/5 = 2.4

(b) If we add one additional 3 to the bag, there are now 9 slips with a 3 on them and 1 slip with a 5 on it. The probability of drawing a 3 is now 9/10 and the probability of drawing a 5 is 1/10. Therefore, the expected value is:

E(X) = (9/10) * 3 + (1/10) * 5 = 27/10 = 2.7

(c) If we add two additional 3's to the bag, there are now 10 slips with a 3 on them and 0 slips with a 5 on them. The probability of drawing a 3 is now 10/10 = 1. Therefore, the expected value is:

E(X) = 1 * 3 = 3

Therefore, the expected value of the number shown when we draw a single slip of paper is 2.4, the expected value if we add one additional 3 to the bag is 2.7, and the expected value if we add two additional 3's to the bag is 3. 

 Mar 7, 2023

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