Solved Problems

A batch of concrete cylinders has a 10% chance ($p=0.10$) of failing a strength test. If 5 cylinders are randomly selected for testing ($n=5$), what is the probability that exactly 1 fails?

Vehicles arrive at a toll booth at an average rate of $\lambda = 3$ per minute. What is the probability of exactly 2 arrivals in a given minute?

The number of potholes on a highway follows a Poisson distribution with an average of 2 potholes per mile ($\lambda = 2$). What is the probability that there are at least 1 pothole in a specific mile?

A robotic welder on a manufacturing line produces a defective joint with a probability of $p = 0.05$. The machine produces joints until a defective one is found, at which point it is shut down for calibration. What is the probability that the machine produces exactly 10 good joints before the first defective one?