Question

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

A glass manufacturer finds that 1 in every 200 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)- (Round to three decimal places as needed.)

Answer 1

We know that the probability that a glass item is warped is 1/200 = 0.005

a.

For the first warped glass item to be the 10th item produced, the first 9 would have to be non warped, and the tenth one warped.

Thus, the probability would be

= 0.995⁹ * 0.005¹

= 0.9558896 * 0.005

= 0.004779448 or 0.4779%

b.

The first warped item is the first, second or third item produced would be

the probability of first warped item being the first item +

the probability of first warped item being the second item +

the probability of first warped item being the third item

= 0.005+ 0.995 * 0.005 + 0.995² * 0.005

= 0.005 (1+ 0.995 + 0.995² )

= 0.005 ( 1 + 0.995 + 0.990025 )

= 0.005 ( 2.985025 )

= 0.01492512 or 1.4925%

c.

This would mean that all the ten items are non warped.

Thus, required probability is

= 0.995¹⁰

= 0.9511101 or 95.1110%