Math Problem Statement

A MACHINE PRODUCES ELECTRONIC COMPONENTS THAT ARE PACKAGED INTO PACKS OF 10. THE PROBABILITY THAT A COMPONENT IS DEFECTIVE IS 0.003, INDEPENDEDNTLY OF ALL OTHER COMPONENTS. FIND THE PROBABILITY THAT AT LEAST ONE OF THE COMPONENTS IN THE PACK IS DEFECTIVE.

tHE MANUFACTURER USES THE FOLLOWING QUALITY CONTROL PROCEDURE TO CHECK LARGE BATCHES OF BOXES:

A PACK OF 10 IS SELECTED AT RANDOM FROM THE BATCH. IF THE PACK CONTAINS ATLEAST ONE DEFECTIVE COMPONENT THEN ANOTHER PACK IS SELECTED FROMT HE SAME BATCH. IF THAT PACK CONTAINS AT LEAST ONE DEFECTIVE COMPONENT, THEN THE WHOLE BATCH IS REJECTED; OTHERWISE THE WHOLE BATCH IS ACCEPTED.

Find the probability that a batch is rejected and suggest a reason why the assumption of independence might not hold.

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Binomial Distribution

Formulas

P(X = 0) = (1 - p)^{10} for 0 defective components
P(at least 1 defective) = 1 - (0.997)^{10}
P(batch rejected) = (1 - (0.997)^{10})^2

Theorems

Binomial Distribution Theorem
Complement Rule

Suitable Grade Level

Grades 10-12