In his doctoral thesis, L. A. Beckel (University of Minnesota, 1982) studied the social behavior of river otters during the mating season. An important role in the bonding process of river otters is very short periods of social grooming. After extensive observations, Dr. Beckel found that one group of river otters under study had a frequency of initiating grooming of approximately 1.7 for every 10 minutes. Suppose that you are observing river otters for 30 minutes. Let r = 0, 1, 2, . . . be a random variable that represents the number of times (in a 30-minute interval) one otter initiates social grooming of another.

(a) Explain why the Poisson distribution would be a good choice for the probability distribution of r. What is l? Write out the formula for the probability distribution of the random variable r.

(b) Find the probabilities that in your 30 minutes of observation, one otter will initiate social grooming four times, ­five times, and six times.

(c) Find the probability that one otter will initiate social grooming four or more times during the 30-minute observation period.

(d) Find the probability that one otter will initiate social grooming fewer than four times during the 30-minute observation period.