Math Problem Statement
The auto parts department of an automotive dealership sends out a mean of 6.7 special orders daily. What is the probability that, for any day, the number of special orders sent out will be exactly 4 ? Round your answer to four decimal places.
Solution
To solve this, we can model the scenario using a Poisson distribution because it describes the probability of a given number of events happening in a fixed interval of time when these events occur with a known constant mean rate.
For a Poisson distribution, the probability of observing exactly events (in this case, 4 special orders) when the average rate is (mean = 6.7 orders) is given by:
where:
- is the base of the natural logarithm (approximately 2.71828).
Step-by-Step Solution
-
Substitute the values into the Poisson formula:
-
Calculate :
-
Calculate :
-
Calculate :
-
Combine all parts to get the probability:
Answer:
The probability that exactly 4 special orders are sent out on a given day is approximately 0.1036.
Would you like further details on the Poisson distribution or additional examples?
Here are 5 related questions to deepen your understanding:
- What is the probability of receiving exactly 5 special orders on a day?
- How would the probability change if the mean increased to 7 orders per day?
- What is the probability of receiving more than 4 special orders on a given day?
- If the mean rate of special orders changes daily, would the Poisson distribution still apply?
- How can the Poisson distribution be used to model events other than special orders?
Tip: The Poisson distribution is particularly useful for events that occur independently and at a constant average rate over time or space.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability
Poisson Distribution
Formulas
Poisson probability formula: P(X = k) = (e^(-λ) * λ^k) / k!
Theorems
Poisson Distribution Theorem
Suitable Grade Level
Grades 11-12
Related Recommendation
Calculate Probability Using Poisson Distribution for Insurance Company Arrivals
Find the Probability of 6 Typographical Errors Using Poisson Distribution
Poisson Probability Distribution with λ = 4.7: Calculating Probabilities
Poisson Distribution: Probability of Abandoned Calls with 2.7 Calls in 5 Minutes
Calculating Probability of More Than Two Errors Using Poisson Distribution