vacuum cleaner company has three machines, each making electric cords.
Machine 1 is the oldest and of all units it produces 1.2% do not meet quality standard (of not being damaged after an abrupt change of voltage). Machine 2 produces 0.8% of bad units, while the newest, machine 3, makes 0.5% of units with unacceptable quality. Of all the cords produced, 20% are made on machine 1, 35% are made on machine 2 and the rest on the machine 3. The cord is used as a part of vacuum cleaner that the company makes. If a customer randomly buys a vacuum cleaner produced by this company, what is the probability that the cord does
not meet the quality standard?
The probability that the cord does not meet the quality standard is 0.8%.
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To find the probability that the cord does not meet the quality standard, we can use the law of total probability.
Let's define the following events:
A: The cord is produced by machine 1
B: The cord is produced by machine 2
C: The cord is produced by machine 3
D: The cord does not meet the quality standard
We are given the following information:
P(D|A) = 0.012 (1.2%)
P(D|B) = 0.008 (0.8%)
P(D|C) = 0.005 (0.5%)
P(A) = 0.20 (20%)
P(B) = 0.35 (35%)
P(C) = 1 - P(A) - P(B) = 1 - 0.20 - 0.35 = 0.45 (45%)
Using the law of total probability, we can calculate the probability that the cord does not meet the quality standard:
P(D) = P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C)
= 0.012 * 0.20 + 0.008 * 0.35 + 0.005 * 0.45
= 0.0024 + 0.0028 + 0.00225
= 0.00745
Therefore, the probability that the cord does not meet the quality standard is 0.00745 or 0.745%.
To calculate the probability that the cord does not meet the quality standard, we need to consider the proportion of cords produced by each machine and their respective failure rates.
Let's denote:
P(M1) = Proportion of cords produced by Machine 1 (oldest machine) = 20%
P(M2) = Proportion of cords produced by Machine 2 = 35%
P(M3) = Proportion of cords produced by Machine 3 (newest machine) = 100% - P(M1) - P(M2) = 45% (as it is stated that the rest of the cords are produced by Machine 3)
F(M1) = Failure rate of cords produced by Machine 1 = 1.2%
F(M2) = Failure rate of cords produced by Machine 2 = 0.8%
F(M3) = Failure rate of cords produced by Machine 3 = 0.5%
To calculate the probability that a randomly bought vacuum cleaner's cord does not meet the quality standard, we need to consider the weighted average of failure rates based on the proportion of cords produced by each machine.
P(cord does not meet the quality standard) = P(M1) * F(M1) + P(M2) * F(M2) + P(M3) * F(M3)
Substituting the values:
P(cord does not meet the quality standard) = (0.2 * 0.012) + (0.35 * 0.008) + (0.45 * 0.005)
Calculating:
P(cord does not meet the quality standard) = 0.0024 + 0.0028 + 0.00225
= 0.00745
Therefore, the probability that the cord does not meet the quality standard is 0.00745 (or 0.745%).