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 12th 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 12th 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) =
(round to three decimal places as needed.)

Explanation:

Step1: Identify the probability of success

The probability \( p \) that a glass item is warped is \( p=\frac{1}{200} = 0.005 \). The probability of failure \( q=1 - p=1 - 0.005 = 0.995 \).

Step2: Use the geometric probability formula for the first success in the first, second, or third trial

The geometric probability formula for the first success on the \( k \)-th trial is \( P(X = k)=q^{k - 1}p \). We need to find \( P(X = 1)+P(X = 2)+P(X = 3) \).

• For \( k = 1 \): \( P(X = 1)=q^{0}p=1\times0.005 = 0.005 \)
• For \( k = 2 \): \( P(X = 2)=q^{1}p=0.995\times0.005=0.004975 \)
• For \( k = 3 \): \( P(X = 3)=q^{2}p=(0.995)^{2}\times0.005=0.990025\times0.005 = 0.004950125 \)

Step3: Sum the probabilities

\( P(X\leq3)=P(X = 1)+P(X = 2)+P(X = 3)=0.005+0.004975 + 0.004950125=0.014925125\approx0.015 \)

Answer:

\( 0.015 \)