A beverage plant has three lines (line A, line B and line C) for bottling the beverages. Line A bottles at a rate of 800 per hour, of which 5 percent have short fill; Line B bottles at a rate of 700 per hour, of which 3 percent have short fill; Line C bottles at a rate of 500 per hour, of which 2 percent have short fill. The lines operate for the same number of hours per day. A bottle selected at random from the production is found to have short fill. What are the probabilities that this bottle comes from line A, line B, or line C?

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Asked Oct 25, 2019
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A beverage plant has three lines (line A, line B and line C) for bottling the beverages. Line A bottles at a rate of 800 per hour, of which 5 percent have short fill; Line B bottles at a rate of 700 per hour, of which 3 percent have short fill; Line C bottles at a rate of 500 per hour, of which 2 percent have short fill. The lines operate for the same number of hours per day. A bottle selected at random from the production is found to have short fill. What are the probabilities that this bottle comes from line A, line B, or line C?

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Expert Answer

Step 1

Let A be the event of selecting line A, B be the event of sleeting line B and C be the event of selecting line C.

Since there are three lines in a beverage plant, hence the probability of selecting any line is equally likely and mutually exclusive.

 

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Step 2

It is given that the line A has 5 percent short fill bottles, line B has 3 percent short fill bottles and Line C has 2 percent short fill bottles.

Let S be the event that the selected bottle is short fill, then the priory probability are,

 

5
P
100
3
and P
100
2
B
100
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5 P 100 3 and P 100 2 B 100

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Step 3

The probability that the short fill bo...

P(A) P(S/A)
P(A) P(SA)+P(B) P(S B)+P(C) P(S/C)
P(A S)
5
3八100
5
2
3八100
100
100
0.016
t0.016+0.01+0.006J
0.016
0.032
=0.50
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P(A) P(S/A) P(A) P(SA)+P(B) P(S B)+P(C) P(S/C) P(A S) 5 3八100 5 2 3八100 100 100 0.016 t0.016+0.01+0.006J 0.016 0.032 =0.50

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