A sales firm receives an average of four calls per hour on its toll-free number. For any given hour, find the probability that it will receive exactly eight calls

To find the probability that the sales firm will receive exactly eight calls in a given hour, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space when these events occur with a known constant mean rate.

The formula for the Poisson distribution is:

P(x; λ) = (e^-λ * λ^x) / x!

Where:
- P(x; λ) is the probability of x events occurring when the mean rate is λ.
- e is a mathematical constant equal to approximately 2.71828.
- λ is the average rate (mean) of events per interval.
- x is the actual number of events that result.

In this case, the average rate of calls per hour is λ = 4, and we want to find the probability of receiving exactly eight calls, so x = 8.

Plugging these values into the Poisson distribution formula, we have:

P(8; 4) = (e^-4 * 4^8) / 8!

Now, let's calculate the probability step by step.

Step 1: Calculate e^-λ
e is a mathematical constant approximately equal to 2.71828, and λ = 4.

e^-4 ≈ 0.01831564

Step 2: Calculate 4^8
4^8 = 65,536

Step 3: Calculate 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320

Step 4: Plug in the values in the formula
P(8; 4) = (e^-4 * 4^8) / 8!
P(8; 4) ≈ (0.01831564 * 65,536) / 40,320

Step 5: Calculate the final probability
P(8; 4) ≈ 0.02820

Therefore, the probability that the sales firm will receive exactly eight calls in a given hour is approximately 0.02820 or about 2.82%.

To find the probability that the sales firm will receive exactly eight calls in any given hour, we can use the Poisson distribution formula.

The Poisson distribution is commonly used to model the number of events that occur in a fixed interval of time or space if these events occur with a known constant rate and independently of the time since the last event. In this case, the average number of calls per hour is given as four.

The probability mass function (PMF) of the Poisson distribution is given by the formula:

P(X = k) = (e^(-λ) * λ^k) / k!

Where:
- P(X = k) represents the probability of receiving exactly k calls in a given hour.
- e is the mathematical constant approximately equal to 2.71828.
- λ (lambda) is the average number of calls per hour.
- k is the number of calls received.

In this case, λ (lambda) is equal to four and k is equal to eight. Let's calculate the probability:

P(X = 8) = (e^(-4) * 4^8) / 8!

Calculating this expression will give you the probability that the sales firm will receive exactly eight calls in any given hour.

Try the Poisson Distribution.

Poisson distribution (m = mean):
P(x) = e^(-m) m^x / x!

Values:
x = 8
m = 4

Substitute and calculate.