Find the indicated probabilities using the geometric distribution, the Poisson d

Question

Find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine if the events are unusual. If convenient, use the appropriate probability table or technology to find the probabilities.

A glass manufacturer finds that 1 in every 250 glass items produced is warped. Find the probability that (a) the first warped glass item is the 10th item produced, (b) the first warped item is the first, second, or third item produced, and (c) none of the first 10 glass items produced are defective.

(a) P(the first warped glass item is the 10th item produced)=

(b) P(the first warped item is the first, second, or third item produced=

(c) P(none of the first 10 glass items produced are defective)=

Which of the events are unusual?

Explanation / Answer

(a) P(the first warped glass item is the 10th item produced)

Here p(warped glass) = 1/250 = 0.004

We have to find P(the first warped glass item is the 10th item produced), to calculate this we know that first 9 items are not warped and the thenth item is warped. The probability is a geometric distribution.

so P( First warped item is 10th one) = (1 - 0.004)9 * (0.004) = 0.003858

(b) P(the first warped item is the first, second, or third item produced) = P (first item ) + P(second item) + P( third item) = 0.004 + (0.996) * (0.004) + (0.996)2 (0.004) = 0.01195

(c) P(none of the first 10 glass items produced are defective) = none of the first 10 glass items are defective

= 10C0 * (0.996)10 = 0.9607

Here part (a) is the most unusual that a specific item shall be warped.

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