What is the probability that, on a given day, Rooneys will receive at most two web-based enquiries for holiday packages? What is the probability that the travel agency will receive... What is the probability that, on a given day, Rooneys will receive at most two web-based enquiries for holiday packages? What is the probability that the travel agency will receive more than two web-based enquiries for travel packages on a given day? Read more

Steps to Solve

1. Identify the Distribution Type Rooneys Travel and Tours receives an average of five enquiries per day. This suggests a Poisson distribution with a mean ($\lambda$) of 5.

2. Calculate the Probability for Part (a) To find the probability of receiving at most 2 enquiries, we calculate the sum of the probabilities for receiving 0, 1, and 2 enquiries using the Poisson probability formula:

$$ P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!} $$

For $k = 0, 1, 2$:

• $P(X = 0) = \frac{5^0 e^{-5}}{0!} = e^{-5}$
• $P(X = 1) = \frac{5^1 e^{-5}}{1!} = 5 e^{-5}$
• $P(X = 2) = \frac{5^2 e^{-5}}{2!} = \frac{25 e^{-5}}{2}$

Now, sum these probabilities:

$$ P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2) $$

3. Calculate the Total for Part (a) Compute:

$$ P(X \leq 2) = e^{-5} + 5 e^{-5} + \frac{25 e^{-5}}{2} $$

Combine:

$$ P(X \leq 2) = e^{-5} \left(1 + 5 + \frac{25}{2}\right) = e^{-5} \left(\frac{32}{2}\right) = 16 e^{-5} $$

4. Calculate the Probability for Part (b) To find the probability of receiving more than 2 enquiries, we can use the complement:

$$ P(X > 2) = 1 - P(X \leq 2) $$

Therefore, substitute the result from part (a):

$$ P(X > 2) = 1 - 16 e^{-5} $$

5. Summarize the Results Both results will need numerical evaluation of $e^{-5}$, which is approximately 0.00674.

Thus:

• For Part (a): Plug into the equation to find $P(X \leq 2)$.
• For Part (b): Subtract this from 1 to find $P(X > 2)$.