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

Explanation:

Step1: Identify the distribution

We use the geometric distribution. The probability of success (a warped glass) is \( p=\frac{1}{200} = 0.005\). The geometric probability formula for the first success on the \( k\)-th trial is \( P(X = k)=(1 - p)^{k - 1}p\).

Step2: Substitute values

For \( k = 12\), \( p = 0.005\), so \( 1-p=0.995\). Then \( P(X = 12)=(0.995)^{11}\times0.005\).

First, calculate \( (0.995)^{11}\). Using a calculator, \( 0.995^{11}\approx0.946\). Then multiply by \( 0.005\): \( 0.946\times0.005 = 0.00473\approx0.005\) (rounded to three decimal places). Wait, more accurately, \( 0.995^{11}=e^{11\ln(0.995)}\approx e^{11\times(- 0.0050125)}\approx e^{-0.0551375}\approx0.9463\). Then \( 0.9463\times0.005 = 0.0047315\approx0.005\) (rounded to three decimal places). Wait, actually, let's compute \( (0.995)^{11}\) more precisely:

\( 0.995^{1}=0.995\)

\( 0.995^{2}=0.995\times0.995 = 0.990025\)

\( 0.995^{3}=0.990025\times0.995=0.985074875\)

\( 0.995^{4}=0.985074875\times0.995 = 0.9801494006\)

\( 0.995^{5}=0.9801494006\times0.995=0.9752436536\)

\( 0.995^{6}=0.9752436536\times0.995 = 0.9703574353\)

\( 0.995^{7}=0.9703574353\times0.995=0.9655006481\)

\( 0.995^{8}=0.9655006481\times0.995 = 0.9606721459\)

\( 0.995^{9}=0.9606721459\times0.995=0.9558737852\)

\( 0.995^{10}=0.9558737852\times0.995 = 0.9511044163\)

\( 0.995^{11}=0.9511044163\times0.995=0.9463489942\)

Then \( P(X = 12)=0.9463489942\times0.005 = 0.00473174497\approx0.005\) (rounded to three decimal places). Wait, actually, when rounded to three decimal places, \( 0.00473\approx0.005\)? No, \( 0.00473\) rounded to three decimal places is \( 0.005\)? Wait, \( 0.00473\) is \( 0.005\) when rounded to three decimal places? Wait, the third decimal place is 4, the next digit is 7, so we round up: \( 0.005\). Wait, no: \( 0.00473\) has the first decimal place 0, second 0, third 4, fourth 7. So rounding to three decimal places: look at the fourth digit, 7, which is ≥5, so we round the third digit (4) up to 5. So \( 0.005\).

Answer:

\( 0.005\)