Question 1193174: Jim is a real estate agent who sells large commercial buildings. Because his commission is so large on a single sale, he does not need to sell many buildings to make a good living. History shows that Jim has a record of selling an average of 8.2 large commercial buildings every 275 days.
In a 52-day period, what is the probability that Jim will make no sales? one sale? two or more sales? (Use two decimal places for 𝜆. Round your answers to four decimal places.)
P(0)
P(1)
P(r ≥ 2)
In a 108-day period, what is the probability that Jim will make no sales? two sales? three or more sales? (Use two decimal places for 𝜆. Round your answers to four decimal places.)
P(0)
P(2)
P(r ≥ 3)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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
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