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?

h(x; 4, 0.2)

b(x; 4, 2, 10)

    

nb(x; 4, 2, 10)

b(x; 4, 0.2)

h(x; 4, 2, 10)

nb(x; 4, 0.2)

(b)

What is the probability that you purchase six boxes? (Round your answer to four decimal places.)

.256

(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?

Answer 1

Given Information:

Let X denotes the number of boxes that do not contain a prize until you have obtained four of these prizes.

The probability that a randomly selected box of a certain type of cereal has a particular prize is given to be 0.2.

a) The probability that you purchase x boxes that do not have the desired prize is given by:

.

b) The probability that you purchase six boxes is calculated as:

The probability that you purchase six boxes is 0.0352.

c) The probability that you purchase at most six boxes is calculated as follows:

The probability that you purchase at most six boxes is 0.1208.

The probability that a randomly selected box of a certain type of cereal has a particular prize is 0.2.

The probability that a randomly selected box of a certain type of cereal does not have a particular prize is calculated as:

.

d)

The expected number of boxes without the desired prize purchased is calculated using the mean of negative binomial distribution and the probability that a randomly selected box of a certain type of cereal does not have a particular prize is the probability of success here that is and is given by:

Hence, the expected number of boxes without the desired prize purchased is 16.

The expected number of boxes purchased is given by:

The expected number of boxes purchased is 20.

X N.B (4,0.2) r=4 p=0.2 P(X = x) =x+4–1 C4–1 x 0.24 x (1 – 0.2) P(X = 1) =2+3 C3 x 0.24 x (0.8)"

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q=1-P =1-0.2 = 0.8

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