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?
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 Poisson distribution.
The Poisson distribution is used when we have a fixed average rate of occurrence (in this case, 7 people per hour) and want to find the probability of a specific number of occurrences during a given time period.
To calculate the probability using the Poisson distribution, we need two pieces of information:
1. The average rate of occurrence (λ) - in this case, it is 7 people per hour.
2. The number of occurrences we're interested in (k) - in this case, exactly 7 people.
The formula for the Poisson probability is:
P(k, λ) = (e^(-λ) * λ^k) / k!
Where e is the mathematical constant approximately equal to 2.71828.
We can now plug in the values:
λ = 7 (average rate of occurrence)
k = 7 (number of occurrences)
P(7, 7) = (e^(-7) * 7^7) / 7!
Using a calculator, we can solve this equation.
Now, let's calculate it step by step:
1. Calculate e^(-7) = 0.000911882
2. Calculate 7^7 = 823,543
3. Calculate 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
Now, plug in the values:
P(7, 7) = (0.000911882 * 823,543) / 5,040
Calculating this, we get:
P(7, 7) ≈ 0.149
Therefore, the probability that exactly 7 people will use the teller machine from 3:00-4:00 pm on Friday is approximately 0.149 or 14.9%.