(Solved): At Burnt Mesa Pueblo, in one of the archaeological excavation sites, the artifact density (number o ...

At Burnt Mesa Pueblo, in one of the archaeological excavation sites, the artifact density (number of prehistoric artifacts per 10 liters of sediment) was 1,3. Suppose you are going to dig up and examine 50 liters of sediment at this site. Let. \( r=0,1,2,3, \ldots \) be a random variable that represents the number of prehistoric artifacts found in your 50 liters of sediment (a) Explain why the Poisson distribution would be a good choice for the probability distribution of \( r \). Finding prehistonc artifacts is a rare occurrence. It is reasonable to assume the events are independent. Finding prehistonc artifacts is a common occurrence. It is reasonable to assume the events are independent. Finding prehistonc artifacts is a common occurrence. It is reasonable to assume the events are dependent: Finding prehistoric artifacts is a rare occurrence. It is reasonable to assume the events are dependent. What is \( \lambda \) ? Write out the formula for the probability distribution of the random variable \( r \). (Use \( e, \lambda \), and \( r \) in your answer.) \( P(r)= \) (b) Compute the probabilities that in your 50 liters of sediment you will find two prehistoric artifacts, three prehistoric artifacts, and four prehistoric artifacts. (Round your answers to four decimal places.) \[ \begin{array}{l} P(2)= \\ P(3)= \\ P(4)= \end{array} \] (c) Find the probability that you will find three or more prehistoric artifacts in the so liters of sediment. (Round your answer to four decimal places.) (d) Find the probability that you will find fewer than three prehistone artifacts in the so liters of sediment. (Round your answer to four decimal places.)