Probability : minutes that customers get served

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In summary: P(X<15)=1-e^{-2\cdot 15}$ $P(6\leq X\leq 8)=P(X\leq 8)-P(X\leq 6)=(1-e^{-2\cdot 8})-(1-e^{-2\cdot 6})$
  • #1
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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  • #2
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.
 
  • #3
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:
 
  • #4
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.
 
  • #5
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:
 
  • #6
mathmari said:
Is that correct? :unsure:

I believe so.
 

FAQ: Probability : minutes that customers get served

What is probability?

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

How is probability calculated?

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. This can be represented as a fraction, decimal, or percentage.

How does probability apply to serving customers?

In the context of serving customers, probability can be used to estimate the likelihood of a customer being served within a certain amount of time. This can help businesses plan and improve their service efficiency.

What factors can affect the probability of customers being served within a certain time frame?

The probability of customers being served within a certain time frame can be affected by factors such as the number of staff working, the number of customers waiting, and the efficiency of the service process.

How can probability be used to improve customer service?

By understanding and analyzing the probability of customers being served within a certain time frame, businesses can make informed decisions to improve their service efficiency. This can lead to shorter wait times for customers and overall better customer satisfaction.

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