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 1000 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) = 0.001 (round to three decimal places as needed.)
(b) p(the first warped item is the first, second, or third item produced) = 0.003 (round to three decimal places as needed.)
(c) p(none of the first 10 glass items produced are defective) = \square (round to three decimal places as needed.)

Explanation:

Step1: Identify the probability of success (warped item)

The probability \( p \) that a glass item is warped is \( p = \frac{1}{1000} = 0.001 \). The probability of failure (not warped) is \( q = 1 - p = 1 - 0.001 = 0.999 \).

Step2: Use the binomial probability formula for "none defective" in 10 trials

We want the probability that none of the first 10 items are defective (i.e., all 10 are non - warped). The binomial probability formula is \( P(X = k)=C(n,k)\times p^{k}\times q^{n - k} \), where \( n = 10 \), \( k = 0 \), \( p = 0.001 \), and \( q = 0.999 \). The combination formula \( C(n,k)=\frac{n!}{k!(n - k)!} \), and when \( k = 0 \), \( C(10,0)=\frac{10!}{0!(10 - 0)!}=1 \). So \( P(X = 0)=C(10,0)\times(0.001)^{0}\times(0.999)^{10 - 0} \). Since any non - zero number to the power of 0 is 1, this simplifies to \( P(X = 0)=1\times1\times(0.999)^{10} \).

Step3: Calculate \( (0.999)^{10} \)

Using a calculator, \( (0.999)^{10}\approx0.990045 \) (rounded to six decimal places). Rounding to three decimal places, we get \( 0.990 \).

Answer:

\( 0.990 \)