SOLUTION: How do you apply the Bayes' Theorem? Thank you.
Use Bayes Theorem for the following
In the current second year class of a top ranked US university, all students come from t
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Question 974803: How do you apply the Bayes' Theorem? Thank you.
Use Bayes Theorem for the following
In the current second year class of a top ranked US university, all students come from three states in the US . Kansas, Illinois, and Ohio supply respectively 40%, 50%, and 10% of the students. The failure rate of students is 4%, 2%, and 6% respectively.
What is the probability that a student chosen at random will fail?
Given that a student fails , what is the probability that he or she came form Kansas?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
Let 
denote the event that a randomly chosen student comes from state 
where 
and 1 represents Kansas, 2 represents Illinois, and 3 represents Ohio. Let 
represent the event that a randomly chosen student will fail. We are given the following infomation:
\ =\ 0.4\ \ \ )
\ =\ 0.5 \ \ \ )
\ =\ 0.1)
If a student is from Kansas, then the probability that student will fail is 0.04. That is to say: \ =\ 0.04)
. By the same analysis, we have, overall:
\ =\ 0.04\ \ \ )
\ =\ 0.02 \ \ \ )
\ =\ 0.06)
To answer the first question, 
we use the following:
\ =\ \sum_{i=1}^3\, P(B\ |\ A_i)\,\cdot\,P(A_i)\ =\ P(B\ |\ A_1)\,\cdot\,P(A_1)\ +\ P(B\ |\ A_2)\,\cdot\,P(A_2)\ +\ P(B\ |\ A_3)\,\cdot\,P(A_3))
The arithmetic is left as an exercise for the student.
Second question: Given that has occurred, we then want to calculate the conditional probability of 
. By Bayes' Theorem:
\ =\ \frac{P(B\ |\ A_1)\,\cdot\,P(A_1)}{P(B)})
The rest is arithmetic.
John
My calculator said it, I believe it, that settles it
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