SOLUTION: A customer-service supervisor regularly conducts a survey of customer satisfaction. The results of the latest survey indicate that 8% of customers were not satisfie

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Question 1197392: A customer-service supervisor regularly conducts a survey of customer satisfaction. The results of the latest survey indicate that 8% of customers were not satisfied with the service they received at their last visit to the store. Of those who are not satisfied, only 22% return to the store within a year. Of those who are satisfied, 64% return within a year. A customer has just entered the store. In response to your question, he informs you that it is less than 1 year since his last visit to the store. What is the probability that he was satisfied with the service he received?
Found 3 solutions by ikleyn, ewatrrr, math_tutor2020:
Answer by ikleyn(52810)   (Show Source): You can put this solution on YOUR website!
.
A customer-service supervisor regularly conducts a survey of customer satisfaction.
The results of the latest survey indicate that 8% of customers were not satisfied
with the service they received at their last visit to the store.
Of those who are not satisfied, only 22% return to the store within a year.
Of those who are satisfied, 64% return within a year.
A customer has just entered the store. In response to your question, he informs you
that it is less than 1 year since his last visit to the store.
What is the probability that he was satisfied with the service he received?
~~~~~~~~~~~~~~~~~ 

They want you calculate the conditional probability

    P(a customer was satisfied with the service he received | given that he returned within a year)

This probability is the fraction, whose denominator is the number of customers
that returned within a year, while the numerator is the number of customers
that were satisfied AND returned within a year.


The denominator is equal to  0.08*0.22 + (1-0.08)*0.64 = 0.6064.


The numerator is (1-0.08)*0.64 = 0.5888.


The probability is this fraction/(value)  

    P =  =  = 0.9710  (rounded).

Solved.



Answer by ewatrrr(24785)   (Show Source): You can put this solution on YOUR website!
8% Unsatisfied(22% return in one year), 92% satisfied(64% return in one year)
P(satisfied | returned in one year)
P(A|B) = P(A and B)/P(B)
P =
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

The other tutors approach this problem using conditional probability, and the law of total probability.
Such methods are fine if you understand how the formula works. And it's useful for exam settings where time is important.

I'll approach this using a different method.

Let's say 10,000 people were surveyed.
Feel free to change this number to any positive integer you want.
I selected this large value so that the resulting values in the table below are all integers.

We're told that "8% of customers were not satisfied"
8% of 10,000 = 0.08*10,000 = 800 people weren't satisfied.
This means 10,000 - 800 = 9200 people were satisfied.

Another fact is "Of those who are not satisfied, only 22% return to the store within a year"
Of the 800 unsatisfied people, 0.22*800 = 176 return to the store within a year, and 800-176 = 624 do not return within a year.

We're also told that "Of those who are satisfied, 64% return within a year"
64% of those 9200 people satisfied = 0.64*9200 = 5888 satisfied people return within a year. The other 9200-5888 = 3312 satisfied customers do not return within a year.

Here's a table to keep track of all the values mentioned so far.
SatisfiedNot SatisfiedTotal
Return58881766064
No return33126243936
Total920080010,000
"Return" of course refers to "Return within 1 year".

Let's move onto the part that says "A customer has just entered the store. In response to your question, he informs you that it is less than 1 year since his last visit to the store"

This means we focus entirely on the "Return" row.
There are 5888+176 = 6064 people who returned within a year of their last visit.

Of those 6064 people, 5888 were satisfied.
Divide those values to get
5888/6064 = 0.97097625329816 approximately
which when rounding to five decimal places we get 0.97098
However, if your teacher needs you to round to some other level of precision, then be sure to do so.

Normally I would go with the fraction form since it's exact and avoids rounding error.
But it's a bit tricky to visualize 5888/6064 in my opinion.
It's much easier to grasp what 0.97098 means (about 97.098% of the people who return within a year are satisfied customers).


Answer: Approximately 0.97098
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