On the average, 6 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, at most 4 people will use the machine?

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YIUYI

To find the probability that at most 4 people will use the machine from 3:00-4:00pm on Friday, we need to calculate the probability for each possible number of people using the machine: 0, 1, 2, 3, or 4. We will then sum up these individual probabilities.

Step 1: Calculate the average number of people that use the machine in the given time frame.
If 6 people per hour use the machine on average, then in a 1-hour time frame, we can expect an average of 6 people to use the machine.

Step 2: Calculate the probability of each scenario.
We need to calculate the probability for 0, 1, 2, 3, and 4 people using the machine in the given time frame.

To calculate these probabilities, we can use the Poisson distribution formula:
P(x; λ) = (e^(-λ) * λ^x) / x!, where x is the number of occurrences, and λ is the average number of occurrences.

Let's calculate the probabilities for each scenario:

P(0; 6) = (e^(-6) * 6^0) / 0! = (e^(-6) * 1) / 1 ≈ 0.00248
P(1; 6) = (e^(-6) * 6^1) / 1! = (e^(-6) * 6) ≈ 0.01488
P(2; 6) = (e^(-6) * 6^2) / 2! = (e^(-6) * 36) / 2 ≈ 0.04465
P(3; 6) = (e^(-6) * 6^3) / 3! = (e^(-6) * 216) / 6 ≈ 0.08930
P(4; 6) = (e^(-6) * 6^4) / 4! = (e^(-6) * 1296) / 24 ≈ 0.11162

Step 3: Sum up the probabilities.
To find the probability that at most 4 people will use the machine, we need to sum up the probabilities of 0, 1, 2, 3, and 4 people using the machine.

P(at most 4) = P(0) + P(1) + P(2) + P(3) + P(4)
≈ 0.00248 + 0.01488 + 0.04465 + 0.08930 + 0.11162
≈ 0.26293

Therefore, the probability that, from 3:00-4:00pm on Friday, at most 4 people will use the machine is approximately 0.26293 or 26.293%.