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 use the Poisson approximation here, we first calculate the expected number of failures in a sample of 90 cars. This is given by \( \lambda = n \cdot p = 90 \cdot 0.03 = 2.7 \). The probability of observing exactly \( k = 3 \) failures can be calculated using the Poisson formula: \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] Substituting the values, we have: \[ P(X = 3) = \frac{e^{-2.7} \cdot 2.7^3}{3!} = \frac{e^{-2.7} \cdot 19.713}{6} \] Now, calculating \( e^{-2.7} \approx 0.0672 \): \[ P(X = 3) \approx \frac{0.0672 \cdot 19.713}{6} \approx 0.2207 \] So, the probability that exactly 3 out of 90 cars will fail the emissions inspection is approximately \( 0.2207 \), or \( 22.07\% \). Enjoy the drive! Curious fact: The Poisson distribution is wonderful for modeling rare events! It’s commonly used in fields like telecommunications, traffic flow analysis, and even in epidemiology to predict occurrences of rare diseases. Just think about it next time you see those funky statistics! For the gearheads: knowing how emissions work in a car can help prevent failures during inspections. Simple maintenance, like checking your engine light or replacing old spark plugs, can drastically reduce emissions and help you pass your inspection with flying colors! Keep your car healthy and running clean!