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Question
Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances 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 and yoga?
Solution
Let events E1, E2 and E be the event of benefiting from meditation and yoga, the event of being treated with medicine and the event of having a heart attack respectively, then
`P (E_1) = 1/2 P (E_2) = 1/2 P(E) = 40% = 0.4`
It is said that meditation and yoga reduce the risk of heart attack by 30%.
That means there is a 70% risk of heart attack.
Or `E/E_1` = Meditation and yoga causes heart attack.
`P(E/E_1)` = 0.40 × 7.0 = 0.28
The drug reduces the risk of heart attack by 25%.
That means the risk of heart attack due to the drug is 75%.
∴ `P(E/E_2) = 0.4 × 0.75 = 0.30`
Thus, `P(E_1) = 1/2, P(E_2) = 1/2`
`P(E/E_1) = 0.28, P(E/E_2) = 0.30`
Hence, by Bayes' theorem,
`P(E_1/E) = (P(E_1) xx P((E)/E_1))/(P(E_1) xx P((E)/E_1) + P(E_2) xx P((E)/E_2))`
= `(1/2 xx 0.28)/(1/2 xx 0.28 + 1/2 xx 0.30)`
= `28/(28 + 30)`
= `28/58`
= `14/29`
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