In California, the average number of potholes in any ten-mile stretch of road is 3.6 . What is the probability that in a given ten mile stretch of road there are exactly 3 potholes?

Step 1 ：This problem is about the Poisson distribution, which gives the probability of a given number of events (in this case, potholes) occurring in a fixed interval of time or space (in this case, a ten-mile stretch of road) with a known average rate of occurrence (in this case, 3.6 potholes per ten miles).

Step 2 ：The formula for the Poisson probability is: \(P(k; λ) = λ^k * e^-λ / k!\), where:

Step 3 ：\(P(k; λ)\) is the Poisson probability,

Step 4 ：\(k\) is the actual number of successes that result from the experiment,

Step 5 ：\(λ\) is the mean number of successes that occur in a specified region,

Step 6 ：\(e\) is the number approximately equal to 2.71828 (Euler's number).

Step 7 ：In this case, we want to find the probability of exactly 3 potholes (\(k=3\)) given an average rate of 3.6 potholes per ten miles (\(λ=3.6\)).

Step 8 ：Substituting the values into the formula, we get \(P = 0.212469265750147\).

Step 9 ：Final Answer: The probability that in a given ten mile stretch of road there are exactly 3 potholes is approximately \(\boxed{0.212}\).