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Probability

Question
CBSEENMA12035758

Assume that the chances of a patient having a heart attack is 40%. Assuming that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drug reduces its chance by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation

Solution

Let A, E1, and E2, respectively denote the events that a person has a heart attack, the selected person followed the course of yoga and meditation, and the person adopted the drug prescription.
therefore space space straight P left parenthesis straight A right parenthesis space equals space 0.40
straight P left parenthesis straight E subscript 1 right parenthesis thin space equals space straight P left parenthesis straight E subscript 2 right parenthesis space equals space 1 half
straight P left parenthesis straight A vertical line straight E subscript 1 right parenthesis space equals space 0.40 space cross times space 0.70 space equals space 0.28
straight P left parenthesis straight A vertical line straight E subscript 2 right parenthesis space equals 0.40 space cross times space 0.75 space equals space 0.30
Probability that the patient suffering a heart attack followed a course of meditation and yoga = straight P left parenthesis straight E subscript 1 vertical line straight A right parenthesis
equals space fraction numerator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A vertical line straight E subscript 1 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A vertical line straight E subscript 1 right parenthesis plus straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A vertical line straight E subscript 2 right parenthesis end fraction
equals fraction numerator begin display style 1 half end style cross times 0.28 over denominator begin display style 1 half end style cross times 0.28 plus begin display style 1 half end style cross times 0.30 end fraction
equals fraction numerator 28 over denominator 28 plus 30 end fraction
equals 28 over 58
equals 14 over 29
Now, calculate straight P left parenthesis straight E subscript 2 vertical line straight A right parenthesis.
straight P left parenthesis straight E subscript 2 vertical line straight A right parenthesis space equals space fraction numerator straight P left parenthesis straight E subscript 2 right parenthesis thin space straight P left parenthesis straight A vertical line straight E subscript 2 right parenthesis over denominator straight P left parenthesis straight E subscript 1 right parenthesis thin space straight P left parenthesis straight A vertical line straight E subscript 1 right parenthesis plus straight P left parenthesis straight E subscript 2 right parenthesis space straight P left parenthesis straight A vertical line straight E subscript 2 right parenthesis end fraction
space space space space space space space space equals fraction numerator begin display style 1 half end style cross times 0.30 over denominator begin display style 1 half end style cross times 0.28 plus begin display style 1 half end style cross times 0.30 end fraction
space space space space space space space space space space space equals fraction numerator 30 over denominator 28 plus 30 end fraction
space space space space space space space space space space space space equals 30 over 58
space space space space space space space space space space space space equals 15 over 29
Since straight P left parenthesis straight E subscript 1 vertical line straight A right parenthesis space less than thin space straight P left parenthesis straight E subscript 2 vertical line straight A right parenthesis , the course of yoga and meditation is more beneficial for a patient. 

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