a. What is the probability that, on a given day, Rooneys will receive at most two web-based enquiries for holiday packages? b. What is the probability that the travel agency will r... a. What is the probability that, on a given day, Rooneys will receive at most two web-based enquiries for holiday packages? b. What is the probability that the travel agency will receive more than two web-based enquiries for travel packages on a given day?
Understand the Problem
The question involves calculating probabilities related to a Poisson distribution, as the agency receives an average of five web-based enquiries per day. The first part asks for the probability of receiving at most two enquiries, while the second part asks for the probability of receiving more than two enquiries on a given day.
Answer
a. Approximately $0.1247$. b. Approximately $0.8753$.
Answer for screen readers
a. The probability that, on a given day, Rooneys will receive at most 2 web-based enquiries is approximately $0.1247$.
b. The probability that the travel agency will receive more than 2 web-based enquiries is approximately $0.8753$.
Steps to Solve
- Identify the Poisson Distribution Parameters
The average number of enquiries per day, denoted as $\lambda$, is 5.
- Probability of Receiving at Most 2 Enquiries
We will calculate the probability of receiving at most 2 enquiries using the Poisson probability formula: $$ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} $$
For at most 2 enquiries ($k = 0, 1, 2$):
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For $k = 0$: $$ P(X = 0) = \frac{e^{-5} 5^0}{0!} $$
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For $k = 1$: $$ P(X = 1) = \frac{e^{-5} 5^1}{1!} $$
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For $k = 2$: $$ P(X = 2) = \frac{e^{-5} 5^2}{2!} $$
The total probability for at most 2 is: $$ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) $$
- Probability of Receiving More than 2 Enquiries
To find the probability of receiving more than 2 enquiries, we can use the complement rule: $$ P(X > 2) = 1 - P(X \leq 2) $$
Now, substitute the previously computed result to find $P(X > 2)$.
a. The probability that, on a given day, Rooneys will receive at most 2 web-based enquiries is approximately $0.1247$.
b. The probability that the travel agency will receive more than 2 web-based enquiries is approximately $0.8753$.
More Information
The Poisson distribution is commonly used for modeling the number of events occurring within a fixed interval of time or space. It is defined by its mean, which in this case is the average number of enquiries received per day.
Tips
- Not correctly applying the factorial in the Poisson formula, particularly for larger values of $k$.
- Forgetting to calculate probabilities for all values required when determining cumulative probabilities like $P(X \leq 2)$.
- Misapplying the complement rule when calculating probabilities for greater than conditions.
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