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%.