Question 974803
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Let *[tex \Large A_i] denote the event that a randomly chosen student comes from state *[tex \Large i] where *[tex \Large i\ \in\ \{1, 2, 3\}] and 1 represents Kansas, 2 represents Illinois, and 3 represents Ohio.  Let *[tex \Large B] represent the event that a randomly chosen student will fail. We are given the following infomation:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(A_1)\ =\ 0.4\ \ \ ]*[tex \LARGE P(A_2)\ =\ 0.5 \ \ \ ]*[tex \LARGE P(A_3)\ =\ 0.1]


If a student is from Kansas, then the probability that student will fail is 0.04. That is to say:  *[tex \Large P(B\ |\ A_1)\ =\ 0.04].  By the same analysis, we have, overall:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(B\ |\ A_1)\ =\ 0.04\ \ \ ]*[tex \LARGE P(B\ |\ A_2)\ =\ 0.02 \ \ \ ]*[tex \LARGE P(B\ |\ A_3)\ =\ 0.06]


To answer the first question, *[tex \Large B\ =\ ?] we use the following:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(B)\ =\ \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 *[tex \Large B] has occurred, we then want to calculate the conditional probability of *[tex \Large A_1].  By Bayes' Theorem:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ P(A_1\ |\ B)\ =\ \frac{P(B\ |\ A_1)\,\cdot\,P(A_1)}{P(B)}]


The rest is arithmetic.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it

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