Question 1193174
Certainly, let's calculate the probabilities of Jim making different numbers of sales within the given time periods.

**1. Calculate the Average Sales Rate per Day**

* Average Sales per Day = (8.2 sales) / (275 days) 
* Average Sales Rate (λ) ≈ 0.03 sales/day 

**2. Calculate the Expected Number of Sales for Each Period**

* **52-day period:**
    * λ (52 days) = 0.03 sales/day * 52 days = 1.56 sales

* **108-day period:**
    * λ (108 days) = 0.03 sales/day * 108 days = 3.24 sales

**3. Use the Poisson Distribution**

The Poisson distribution is suitable for modeling the number of events (sales) occurring within a specific time interval, given the average rate of occurrence. 

* **Probability Mass Function (PMF) of Poisson Distribution:**

   P(X = k) = (e^(-λ) * λ^k) / k! 

   where:
      * X is the random variable representing the number of sales
      * k is the number of sales (0, 1, 2, etc.)
      * λ is the average rate of sales within the time period
      * e is the base of the natural logarithm (approximately 2.71828)
      * k! is the factorial of k

**4. Calculate Probabilities for 52-day Period**

* **P(0 sales):**
   P(X = 0) = (e^(-1.56) * 1.56^0) / 0! 
   P(X = 0) ≈ 0.2121

* **P(1 sale):**
   P(X = 1) = (e^(-1.56) * 1.56^1) / 1! 
   P(X = 1) ≈ 0.3289

* **P(2 or more sales):**
   P(X ≥ 2) = 1 - P(X = 0) - P(X = 1) 
   P(X ≥ 2) = 1 - 0.2121 - 0.3289 
   P(X ≥ 2) ≈ 0.4589

**5. Calculate Probabilities for 108-day Period**

* **P(0 sales):**
   P(X = 0) = (e^(-3.24) * 3.24^0) / 0! 
   P(X = 0) ≈ 0.0399

* **P(2 sales):**
   P(X = 2) = (e^(-3.24) * 3.24^2) / 2! 
   P(X = 2) ≈ 0.2071

* **P(3 or more sales):**
   P(X ≥ 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2) 
   P(X ≥ 3) = 1 - 0.0399 - (e^(-3.24) * 3.24^1) / 1! - 0.2071 
   P(X ≥ 3) ≈ 0.6243 

**In summary:**

**52-day period:**

* P(0 sales): 0.2121
* P(1 sale): 0.3289
* P(2 or more sales): 0.4589

**108-day period:**

* P(0 sales): 0.0399
* P(2 sales): 0.2071
* P(3 or more sales): 0.6243