Question 9 - of 20 Step 1 of 1
Cars enter a car wash at a mean rate of 2 cars per half an hour. What is the probability that, in any hour, at least 3 cars will enter the car wash? Round your answer to four decimal places.

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Step 1 ：This problem involves a Poisson distribution. The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Step 2 ：The mean rate of cars entering the car wash is 2 cars per half an hour, so it's 4 cars per hour. We want to find the probability that at least 3 cars will enter the car wash in any hour.

Step 3 ：The formula for the Poisson distribution is: $P(X=k)=\frac{{\lambda }^k\ast e^{-\lambda }}{k!}$ where: $P(X=k)$ is the probability of k events in the interval, $\lambda$ is the mean rate of value, e is the base of the natural logarithm, and $k!$ is the factorial of k.

Step 4 ：However, since we want to find the probability of at least 3 cars, we need to find the probability of 0, 1, and 2 cars and subtract these from 1.

Step 5 ：Let's calculate the probabilities: $p_0=0.0183$, $p_1=0.0733$, and $p_2=0.1465$.

Step 6 ：Finally, we calculate the probability of at least 3 cars entering the car wash in any hour by subtracting the sum of the probabilities of 0, 1, and 2 cars from 1: $p_{\text{at least 3}}=1-(p_0+p_1+p_2)=0.7619$.

Step 7 ：Final Answer: The probability that, in any hour, at least 3 cars will enter the car wash is $\boxed{{\displaystyle 0.7619}}$.