This problem follows a Poisson distribution, since the number of calls per hour is given as an average rate of occurrence. The Poisson distribution is commonly used to model the number of events occurring within a fixed interval of time or space, when those events occur independently of each other.

Step 1: Define parameters

• The average number of calls per hour ($\lambda$) is 2.
• We are asked to find the probability of receiving more than 3 calls in an hour, i.e., $P(X > 3)$, where $X$ is the random variable representing the number of calls.

Step 2: Poisson probability formula

The Poisson probability mass function (PMF) is given by:

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

Where:

• $k$ is the number of events (calls),
• $\lambda$ is the average number of events (calls per hour),
• $e$ is Euler's number (approximately 2.71828).

We need to find $P(X > 3)$, which is:

$P(X > 3) = 1 - P(X \leq 3)$

Step 3: Calculate $P(X \leq 3)$

To calculate $P(X \leq 3)$, we sum the probabilities for $X = 0, 1, 2, 3$:

$P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)$

Compute each term:

$P(X = 0) = \frac{2^0 e^{-2}}{0!} = \frac{1 \times e^{-2}}{1} = e^{-2} \approx 0.1353$ $P(X = 1) = \frac{2^1 e^{-2}}{1!} = \frac{2 \times e^{-2}}{1} = 2e^{-2} \approx 0.2707$ $P(X = 2) = \frac{2^2 e^{-2}}{2!} = \frac{4 \times e^{-2}}{2} = 2e^{-2} \approx 0.2707$ $P(X = 3) = \frac{2^3 e^{-2}}{3!} = \frac{8 \times e^{-2}}{6} = \frac{4}{3} e^{-2} \approx 0.1805$

Add these up:

$P(X \leq 3) \approx 0.1353 + 0.2707 + 0.2707 + 0.1805 = 0.8572$

Step 4: Calculate $P(X > 3)$

Now we can find $P(X > 3)$:

$P(X > 3) = 1 - P(X \leq 3) = 1 - 0.8572 = 0.1428$

Final Answer:

The probability that there will be more than 3 calls in a randomly selected hour is approximately 0.143 (rounded to the nearest thousandth).

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Would you like a more detailed explanation of any step or another example? Here are some related questions:

1. How would you calculate the probability for $X \leq 2$ in a Poisson distribution?
2. How does the Poisson distribution compare to other distributions like the binomial distribution?
3. What happens to the probability if the average rate of calls ($\lambda$) increases?
4. How would you use the Poisson distribution to model a situation with a smaller time interval, say 30 minutes?
5. Can you explain the relationship between the Poisson and exponential distributions?

Tip: When dealing with the Poisson distribution, remember that the mean and variance are both equal to $\lambda$.