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

Explanation:

Step1: Identify the probability of success (p)

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

Step2: Use the geometric probability formula for each case and sum them

For a geometric distribution, the probability that the first success (warped item) occurs 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 \( k = 1 \): \( P(X = 1) = q^{0}p = 1 \times 0.005 = 0.005 \)
• For \( k = 2 \): \( P(X = 2) = q^{1}p = 0.995 \times 0.005 \approx 0.004975 \)
• For \( k = 3 \): \( P(X = 3) = q^{2}p = (0.995)^{2} \times 0.005 \approx 0.990025 \times 0.005 \approx 0.004950125 \)

Now, sum these probabilities:
\( P(X = 1) + P(X = 2) + P(X = 3) \approx 0.005 + 0.004975 + 0.004950125 \approx 0.014925125 \)

Step3: Round to three decimal places

Rounding \( 0.014925125 \) to three decimal places gives \( 0.015 \).

Answer:

\( 0.015 \)