Probability : minutes that customers get served

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Discussion Overview

The discussion revolves around calculating probabilities related to customer service times at a department store, specifically using the Poisson and exponential distributions. Participants explore the implications of a service rate of 30 customers per hour and how it translates to wait times in minutes.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Exploratory

Main Points Raised

  • One participant questions whether the service time can be modeled using a Poisson distribution with a rate of $\lambda=30$ and seeks clarification on the use of minutes instead of the number of customers.
  • Another participant asserts that the average service time of 30 people per hour corresponds to an exponential distribution with a mean of 2 minutes per person, suggesting that probabilities can be calculated based on this distribution.
  • There is a discussion about the interpretation of the term "need" in the context of calculating probabilities for service times between 6 and 8 minutes.
  • Several participants confirm that the time between Poisson arrivals follows an exponential distribution, linking the service rate to the properties of these distributions.
  • A participant proposes specific probability calculations for being served in less than 15 minutes and for the time interval of 6 to 8 minutes, seeking validation of their approach.
  • Another participant expresses uncertainty about the correctness of the proposed calculations but indicates agreement with the underlying reasoning.

Areas of Agreement / Disagreement

Participants generally agree on the application of the exponential distribution to the problem, but there remains some uncertainty regarding the interpretation of terms and the correctness of specific probability calculations.

Contextual Notes

Participants have not fully resolved the implications of using "need" in the context of probability calculations, and there is some ambiguity in the proposed calculations that has not been clarified.

mathmari
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Hey! :giggle:

The average time of customer service at the cash registers of a department store is $30$ people per hour. If a new customer arrives at the checkout, then calculate the probabilities:

a) to be served in less than 15 minutes

b) to need to be served from 6 to 8 minutes
a) Do we have Poisson disrtibution with $\lambda=30$ ?
But then we consider "minutes" instead of "number of customers".
Could you give me a hint ?

:unsure:
 
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a) 30 people per hour is 2 minutes per person. The wait time is exponentially distributed with this mean. Evaluate the probability that the wait time is less than 15 min given this distribution.

b) I'm not sure what the word "need" means here. If they mean find the probability the customer is served in 6-8 minutes it's just the CDF of the above mentioned distribution evaluated at 8 minutes same evaluated at 6 minutes.
 
romsek said:
a) 30 people per hour is 2 minutes per person. The wait time is exponentially distributed with this mean. Evaluate the probability that the wait time is less than 15 min given this distribution.

b) I'm not sure what the word "need" means here. If they mean find the probability the customer is served in 6-8 minutes it's just the CDF of the above mentioned distribution evaluated at 8 minutes same evaluated at 6 minutes.

How do we know that we have an exponential distribution? :unsure:
 
mathmari said:
How do we know that we have an exponential distribution? :unsure:

It's a property of the Poisson/Exponential distributions.

30 people an hr implies a Poisson distribution on the number of arrivals during a given period.

It's just a property that the time between Poisson arrivals has an exponential distribution.
 
romsek said:
It's a property of the Poisson/Exponential distributions.

30 people an hr implies a Poisson distribution on the number of arrivals during a given period.

It's just a property that the time between Poisson arrivals has an exponential distribution.

Ah ok!

So do we have the following?

a) $P(X<15)=1-e^{-2\cdot 15}$

b) $P(6\leq X\leq 8)=P(X\leq 8)-P(X\leq 6)=(1-e^{-2\cdot 8})-(1-e^{-2\cdot 6})$

Is that correct? :unsure:
 
mathmari said:
Is that correct? :unsure:

I believe so.
 

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