Question

Judging from recent experience, 5% of worm gears produced by an automatic, high-speed machine are defective. What is the probability that out of six gears selected at random, none of them will be defective ?

Answer 1

Let X denote the number of defectives out of six gears selected at random.

It is given that 5% of worm gears produced by an automatic, high-speed machine are defective.

Hence, p = 0.05

We can see that X follows Binomial distribution with parameters n = 6 and p = 0.05

Probability that out of 6 gears selected, none of them will be defective =

= (1 - 0.05)⁶ = 0.95⁶ = 0.73509

Another approach
We are given the probability of defective gear is 0.05

Probability that the randomly selected gear is non-defective is 0.95

Let X_i denote the effectiveness of i^th gear

X_i = 1 if gear is defective P[X_i=1] = 0.05

X_i = 0 if gear is defective P[X_i=0] = 0.95

Suppose 6 gears are selected at random, then we can assume that they all are independently each other.

Probability that none of them are defective = P[ X₁=0,X₂=0,X₃=0,X₄=0,X₅=0,X₆=0]

= P[X₁=0]* P[X₂=0] *P[X₃=0] *P[X₄=0] *P[X₅=0] *P[X₆=0] = 0.95⁶ = 0.73509

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