17. (20 points) An urn contains 20 balls, 6 of which are white white or not The

Question

17. (20 points) An urn contains 20 balls, 6 of which are white white or not The ball is are drawn in succession and we record whether the ba replaced with each draw before the next new draw. is white or not white. 5 form labe Explain the elements of the mathematical formula. a. Write this down in omial distribution form labeling the elements, n-?, p/ the mathematical formula. b. C. What are th e mean and standard deviation of this distribution? d. Wh at is the probability What is the probability that at least one ball is white? of that exactly 5 of the balls drawn are white? e.

Explanation / Answer

a) n = 8

p = 6/20 = 0.3

P(X = x) = 8Cx * 0.3x * (1 - 0.3)8-x

b) 8Cx is number of ways to choose x success. 0.3x is probability of x success. (1 - 0.3)8-x is probability of 8-x failure

c) Mean = n * p = 8 * 0.3 = 2.4

Standard deviation = sqrt(n * p * (1 - p)) = sqrt(8 * 0.3 * 0.7) = 1.3

d) P(X = 5) = 8C5 * 0.35 * 0.73 = 0.0467

e) P(X > 1) = 1 - P(X = 0) = 1 - 8C0 * 0.30 * 0.78 = 1 - 0.0576 = 0.9424

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