14. Emission Inspection Failures If \( 3 \% \) of all cars fail the emissions inspection, find the probability that in a sample of 90 cars, 3 will fail. Use the Poisson approximation.

To find the probability that in a sample of 90 cars, 3 will fail the emissions inspection, we can use the Poisson approximation. Given: - The probability of a car failing the emissions inspection is \(3\%\), which is \(0.03\). - The sample size is 90 cars. The Poisson distribution is given by the formula: \[ P(X = k) = \frac{e^{-\lambda} \cdot \lambda^k}{k!} \] where: - \( P(X = k) \) is the probability of \( k \) events occurring. - \( e \) is the base of the natural logarithm. - \( \lambda \) is the average rate of events occurring. In this case, we want to find the probability that 3 cars will fail the emissions inspection. So, we set \( k = 3 \). First, we need to calculate the average rate of cars failing the emissions inspection, which is \( \lambda \). This can be calculated as: \[ \lambda = n \cdot p \] where: - \( n \) is the sample size (90 cars). - \( p \) is the probability of a car failing the emissions inspection (0.03). Now, we can substitute the values into the Poisson distribution formula to find the probability that 3 cars will fail the emissions inspection. Calculate the value by following steps: - step0: Calculate: \(90\times 0.03\) - step1: Multiply the numbers: \(2.7\) The average rate of cars failing the emissions inspection is \( \lambda = 2.7 \). Now, we can substitute the values into the Poisson distribution formula to find the probability that 3 cars will fail the emissions inspection. Simplify the expression by following steps: - step0: Solution: \(\frac{e^{-2.7}\left(2.7^{3}\right)}{3!}\) - step1: Evaluate: \(\frac{e^{-2.7}\times 2.7^{3}}{3!}\) - step2: Convert the expressions: \(\frac{e^{-\frac{27}{10}}\times 2.7^{3}}{3!}\) - step3: Convert the expressions: \(\frac{e^{-\frac{27}{10}}\left(\frac{27}{10}\right)^{3}}{3!}\) - step4: Multiply the numbers: \(\frac{\frac{27^{3}}{1000e^{\frac{27}{10}}}}{3!}\) - step5: Multiply by the reciprocal: \(\frac{27^{3}}{1000e^{\frac{27}{10}}}\times \frac{1}{3!}\) - step6: Rewrite the expression: \(\frac{3^{9}}{1000e^{\frac{27}{10}}}\times \frac{1}{3\times 2!}\) - step7: Reduce the fraction: \(\frac{3^{8}}{1000e^{\frac{27}{10}}}\times \frac{1}{2!}\) - step8: Multiply the terms: \(\frac{3^{8}}{1000e^{\frac{27}{10}}\times 2!}\) - step9: Multiply the terms: \(\frac{3^{8}}{2000e^{\frac{27}{10}}}\) - step10: Evaluate the power: \(\frac{6561}{2000e^{\frac{27}{10}}}\) - step11: Multiply by the Conjugate: \(\frac{6561e^{\frac{3}{10}}}{2000e^{\frac{27}{10}}\times e^{\frac{3}{10}}}\) - step12: Multiply the numbers: \(\frac{6561e^{\frac{3}{10}}}{2000e^{3}}\) The probability that in a sample of 90 cars, 3 will fail the emissions inspection is approximately 0.220468.