Question

# An automated machine for a soft drink company is calibrated to fill cans with 12-ounces of...

An automated machine for a soft drink company is calibrated to fill cans with 12-ounces of soda. Occasionally, the machine drifts out of calibration and will not fill the cans with the correct amount of soda. Suppose the machine goes out-of-calibration during a batch run of 12,000. Technicians on site indicate that (about) 600 cans were underfilled before repairs could correct the problem. Assume: x ≡ number of under-filled cans of soda in a randomly selected six-pack. S ≡ a can selected at random is under-filled. Use pictures, diagrams and correct statistical notation to answer the questions that follow:

a) What is the probability of success and the probability of failure for this problem?

b) Use the formula ( ? ? ) = ?! ?!∙(?−?)! to calculate the number of ways that 3-cans of soda can be placed in a sample of size 6 (sixpack) if the order does not matter.

c) The probability mass function for this random variable is given by: ?(? = ?) = ( ? ? ) ∙ ? ? ∙ (1− ?) ?−?. Use the probability mass function to find the probability that a six-pack contains exactly 3-cans of soda that were under-filled.

d) Find the probability that 3 or more cans of soda are underfilled in a randomly selected six-pack.

(a)

the probability of success = p = 600/12000 = 0.05

the probability of failure = 1 - p =0.95

(b)

the number of ways that 3-cans of soda can be placed in a sample of size 6 (sixpack) if the order does not matter is given by:

So,

20

(c)

So,

0.0021

(d)

P(X3)= P(X = 3) + P(X=4) + P(X=5) + P(X=6)

So, substituting, we get:

P(X3)= P(X = 3) + P(X=4) + P(X=5) + P(X=6) = 0.0022

So,

0.0022

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