On the average, 7 people per hour use an express teller machine situated inside a commercial complex. What is the probability that, from 3:00-4:00pm on Friday, exactly 7 people will use the teller machine?

0.0231

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To find the probability that exactly 7 people will use the teller machine from 3:00-4:00pm on Friday, we need to use the concept of a Poisson distribution. The Poisson distribution is commonly used for situations where events occur randomly over time or space.

The Poisson distribution requires two parameters: the average rate of occurrence (in this case, the average number of people using the teller machine per hour) and the desired number of occurrences (in this case, exactly 7 people).

Step 1: Determine the average rate of occurrence. Given that 7 people use the teller machine on average per hour, this is our average rate.

Step 2: Calculate the time interval of interest. In this case, from 3:00-4:00pm, the time interval is 1 hour.

Step 3: Use the Poisson distribution formula to find the probability. The formula is:

P(x; μ) = (e^-μ * μ^x) / x!

Where:
- P(x; μ) is the probability of getting exactly x occurrences given the average rate μ.
- e is the base of the natural logarithm (approximately equal to 2.71828).
- μ is the average rate of occurrence.
- x is the desired number of occurrences.

In this case, we want to find P(x = 7; μ = 7).

P(7; 7) = (e^-7 * 7^7) / 7!

Now, we can calculate the probability.

Step 4: Substitute the values into the equation and solve.

P(7; 7) = (2.71828^-7 * 7^7) / 7!

Using a calculator, we get:

P(7; 7) ≈ 0.1499 or 14.99%

Therefore, the probability that exactly 7 people will use the teller machine from 3:00-4:00pm on Friday is approximately 0.1499 or 14.99%.

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