7. We plant 10 seeds in the ground and water them. After 1 month, one of two pos

Question

7. We plant 10 seeds in the ground and water them. After 1 month, one of two possible things can happen: A) the seed sprouted or B) the seed did not sprout. Let the random variable (RV) X= the number of seeds that have sprouted. Let the probability of a given seed sprouting be equal to 0.8 (i.e., P(A)-0.8 a) What is the average number of seeds that you would expect to have sprouted in 1 month (i.e, the mean(X))? What is the variance()? For var(X). you do not need to calculate the final numeric answer: just set up the summation and populate it with the actual numeric values, What assumption(s) have you made, if any? Show your work. b) What is the probability that exactly 2 seed sprouts? Show your work or justify your answer. Your answer must be numeric but you do not need to simplify it since the answer a very small (i.e., it can be a product of numbers).

Explanation / Answer

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a. The average number of seeds expected to be sprouting in a month = n x P(A) = 10 x 0.8 = 8

If X is the random variable denoting the number of seeds sprouted, we have X = a = n x P(A)

Thus, Var (X) = Var (nP(A)) = n^2 Var (P(A)) = n^2 x (nP(A)(1-P(A))) = 100 x (10 x 0.8 x 0.2)

b. Probability that exactly 2 seeds sprout = Probablity of 2 seeds sprouting x probability of other seeds not sprouting

= (0.8^2) x (0.2^8)

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