An auto parts seller finds that 1 in every 200 parts sold is defective. Use the geometric distribution to find the probability that (a) the first defective part is the tenth part sold, (b) the first defective part is the first, second, or third part sold, and (c) none of the first 10 parts sold are defective.

To solve this problem, we will use the geometric distribution, which models the number of trials until the first success (in this case, finding a defective part). The probability of success (finding a defective part) is given as \( p = \frac{1}{200} \), and the probability of failure (finding a non-defective part) is \( q = 1 - p = \frac{199}{200} \). ### (a) The probability that the first defective part is the tenth part sold The probability that the first defective part is the \( k \)-th part sold is given by the formula: \[ P(X = k) = q^{k-1} \cdot p \] For \( k = 10 \): \[ P(X = 10) = \left(\frac{199}{200}\right)^{9} \cdot \left(\frac{1}{200}\right) \] Now, let's calculate this probability. ### (b) The probability that the first defective part is the first, second, or third part sold We need to find the probability that the first defective part is either the 1st, 2nd, or 3rd part sold. This can be calculated as: \[ P(X = 1) + P(X = 2) + P(X = 3) \] Calculating each term: 1. For \( k = 1 \): \[ P(X = 1) = p = \frac{1}{200} \] 2. For \( k = 2 \): \[ P(X = 2) = \left(\frac{199}{200}\right)^{1} \cdot \left(\frac{1}{200}\right) \] 3. For \( k = 3 \): \[ P(X = 3) = \left(\frac{199}{200}\right)^{2} \cdot \left(\frac{1}{200}\right) \] Now, we will sum these probabilities. ### (c) The probability that none of the first 10 parts sold are defective The probability that none of the first 10 parts are defective is given by: \[ P(\text{none defective}) = q^{10} = \left(\frac{199}{200}\right)^{10} \] Now, let's calculate all these probabilities step by step. I'll perform the calculations for parts (a), (b), and (c) now. Calculate the value by following steps: - step0: Calculate: \(\left(\frac{199}{200}\right)^{10}\) - step1: Use the properties of exponents: \(\frac{199^{10}}{200^{10}}\) Calculate or simplify the expression \( (199/200)^9 * (1/200) \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{199}{200}\right)^{9}\times \frac{1}{200}\) - step1: Evaluate the power: \(\frac{199^{9}}{200^{9}}\times \frac{1}{200}\) - step2: Multiply the fractions: \(\frac{199^{9}}{200^{9}\times 200}\) - step3: Multiply: \(\frac{199^{9}}{200^{10}}\) Calculate or simplify the expression \( (1/200) + (199/200)^1 * (1/200) + (199/200)^2 * (1/200) \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{200}+\left(\frac{199}{200}\right)^{1}\times \frac{1}{200}+\left(\frac{199}{200}\right)^{2}\times \frac{1}{200}\) - step1: Calculate: \(\frac{1}{200}+\frac{199}{200}\times \frac{1}{200}+\left(\frac{199}{200}\right)^{2}\times \frac{1}{200}\) - step2: Multiply the numbers: \(\frac{1}{200}+\frac{199}{40000}+\left(\frac{199}{200}\right)^{2}\times \frac{1}{200}\) - step3: Multiply the numbers: \(\frac{1}{200}+\frac{199}{40000}+\frac{199^{2}}{200^{3}}\) - step4: Reduce fractions to a common denominator: \(\frac{200\times 200}{200\times 200\times 200}+\frac{199\times 200}{40000\times 200}+\frac{199^{2}}{200^{3}}\) - step5: Multiply the terms: \(\frac{200\times 200}{8000000}+\frac{199\times 200}{40000\times 200}+\frac{199^{2}}{200^{3}}\) - step6: Multiply the numbers: \(\frac{200\times 200}{8000000}+\frac{199\times 200}{8000000}+\frac{199^{2}}{200^{3}}\) - step7: Rewrite the expression: \(\frac{200\times 200}{200^{3}}+\frac{199\times 200}{200^{3}}+\frac{199^{2}}{200^{3}}\) - step8: Transform the expression: \(\frac{200\times 200+199\times 200+199^{2}}{200^{3}}\) - step9: Multiply the numbers: \(\frac{40000+199\times 200+199^{2}}{200^{3}}\) - step10: Multiply the numbers: \(\frac{40000+39800+199^{2}}{200^{3}}\) - step11: Add the numbers: \(\frac{119401}{200^{3}}\) Here are the calculated probabilities for each part of the problem: ### (a) The probability that the first defective part is the tenth part sold \[ P(X = 10) = \left(\frac{199}{200}\right)^{9} \cdot \left(\frac{1}{200}\right) \approx 0.004779 \] ### (b) The probability that the first defective part is the first, second, or third part sold \[ P(X = 1) + P(X = 2) + P(X = 3) = \frac{1}{200} + \left(\frac{199}{200}\right)^{1} \cdot \left(\frac{1}{200}\right) + \left(\frac{199}{200}\right)^{2} \cdot \left(\frac{1}{200}\right) \approx 0.014925125 \] ### (c) The probability that none of the first 10 parts sold are defective \[ P(\text{none defective}) = \left(\frac{199}{200}\right)^{10} \approx 0.95111 \] ### Summary of Results: - (a) Probability that the first defective part is the tenth part sold: **0.004779** - (b) Probability that the first defective part is the first, second, or third part sold: **0.014925125** - (c) Probability that none of the first 10 parts sold are defective: **0.95111**