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 200 glass items produced is warped. find the probability that (a) the first warped glass item is the 11th 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 11th item produced) = 0.005 (round to three decimal places as needed.)
(b) p(the first warped item is the first, second, or third item produced) = 0.015 (round to three decimal places as needed.)
(c) p(none of the first 10 glass items produced are defective) = \boxed{} (round to three decimal places as needed.)

Explanation:

Step1: Identify the probability of success

The probability that a glass item is warped, \( p=\frac{1}{200} = 0.005\). The probability of failure (not warped) is \( q = 1 - p=1 - 0.005 = 0.995\).

Step2: Use the geometric probability formula for "none in first n trials"

The probability that none of the first \( n\) trials are successful (here, a "success" is a warped item) in a geometric distribution is given by \( P(X>n)=q^{n}\). For \( n = 10\), we need to calculate \( P(\text{none of the first 10 are defective})=q^{10}\).

Step3: Calculate \( q^{10}\)

Substitute \( q = 0.995\) into the formula: \( (0.995)^{10}\). Using a calculator, \( 0.995^{10}\approx0.951\) (rounded to three decimal places).

Answer:

\( 0.951\)