In Exercises 1–7, consider a grocery store that can process a ...

Question

In Exercises 1–7, consider a grocery store that can process a total of four customers at its checkout counters each minute.

The mean number of arrivals per minute is four. Find the probability that

c. one customer is waiting in line after one minute and no customers are waiting in line after the second minute..

Accepted Answer

Below there, today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Customers arrive at a small coffee shop, according to the Poisson distribution, with a mean rate of 3 customers per minute. The shop can serve a maximum of 3 customers per minute at the counter. What is the probability that exactly 1 customer is waiting after the first minute and that no customers are waiting after the 2nd minute? Awesome. So it appears for this particular problem we're ultimately trying to determine what the probability is that exactly 1 customer is waiting after the first minute and that no customers are waiting after the 2nd minute. So our final answer we're trying to solve for is what is this probability value based on the conditions set to us by the prom itself. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. A is 0.0450. B is 0.2810. C is 0.0711, and finally, D is 0.7924. Awesome. So to quickly recap what exactly we're trying to solve for based on the information that's given to us, we're told, as we should recall, that a coffee shop serves 3 customers per minute. And the mean number of arrivals per minute is also 3. And then we have a Poisson distribution with a mean of lambda is equal to 3. And we're trying to determine the probability that exactly. 4 customers arrive in the first minute, so 1 customer is waiting since the shop can only serve 3 per minute and after minute 2 no customers are waiting means in minute 2 that 2 customers arrive at most and the shop serves at most 2 new arrivals plus 1 from the previous minute. For a total of 3 customers served or fewer. So let X be roughly the Poisson, which is lambda equal to 3. So that would mean that the probability that 4 customers arrive in the first minute, so the probability P of X is equal to 4, will be equal to 3 of 4 multiplied by E to the power of -3. All divided by 4 factoral. Fantastic. So the probability that at most 2 customers arrive in the 2nd minute, we could write this expression as, as we should recall, we could write it as the probability P of X is less than or equal to 2, which is equal to p of X is equal to 0, plus p of X is equal to 1. Awesome, and we could say that this is going to be added to P of X is equal to 2. Which will mean that this whole expression is equal to E to the power of -3 multiplied by parentheses 30 divided by 0 factoral. Plus 3 to 1 divided by 1 factoral. Fantastic, plus 3 to 2 divided by 2 factoral. And don't forget your second parenthesis. So once again, we need to recall a note that in order to find the probability that exactly 1 customer is waiting after the 1st minute and that no customers are waiting after the 2nd minute, we need to multiply the above two probabilities, so we need to take key of. X is equal to 4. And X is less than or equal to 2. is equal to a note that we have an upside down union symbol. It's been flipped upside down, and this is equal to, as we should recall, 3 of 4 multiplied by E to the power of -3. Divided by 4 factoral, multiplied by E to the power of -3, multiplied by parentheses. 3 to the power of 0 divided by 0 factoral, plus 3 the power of 1 divided by 1 factoral. Plus 3 2 divided by 2 factoral. Don't forget your second parenthesis. And when we plug in this expression into our calculator, we will find when we round to 4 decimal places that'll be equal to 0.0711, and that's it. That is our final answer. Hooray, we did it. So looking at our multiple choice answers, the correct answer has to be multiple choice answer letter C, which once again is 0.0711. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Key Concepts

Poisson Distribution

Queueing Theory

Probability Calculation

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