The probability that a randomly selected box of a certain type of cereal has a p

Question

The probability that a randomly selected box of a certain type of cereal has a particular prize is 0.2. Suppose you purchase box after box until you have obtained four of these prizes. (a) What is the probability that you purchase x boxes that do not have the desired prize? nb(x; 4, 0.2) h(x; 4, 0.2) o nb(x; 4, 2, 10) O b(x; 4, 2, 10) O b(x, 4, 0.2) O h(x; 4, 2, 10) (b) What is the probability that you purchase six boxes? (Round your answer to four decimal places.) (c) What is the probability that you purchase at most six boxes? (Round your answer to four decimal places.) (d) How many boxes without the desired prize do you expect to purchase? How many boxes do you expect to purchase?

Explanation / Answer

a)
nb(x, 4, 0.2)

b)
P(X = 6) = nb(6, 4, 0.2)
= 5C3 * 0.2^3 * 0.8^2
= 0.0512

c)
P(X <= 6) = nb(4,4,0.2) + nb(5,4,0.2) + nb(6,4,0.2)
= 0.2^3 + 4C3 * 0.2^3 * 0.8 + 5C3 * 0.2^3 * 0.8^2
= 0.0848

d)
The negative binomial random variable is K, the number of failures before the binomial experiment results in r successes. The mean of K is:

K = r*Q/P = 4*0.8/0.2 = 16

If we define the mean of the negative binomial distribution as the average number of trials required to produce r successes, then the mean is equal to:

= r / P = 4/0.2 = 20

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