Question

Assume that the chances of a patient having a heart attack is 40%. It is also assumed that a meditation and yoga course reduces the risk of heart attack by 30% and prescription of certain drugs 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 medication and yoga?

Answer 1

We will first find the probability of a person having a heart attack. Then, find the probability of adopting yoga and medication and also, find the probability of adopting drug prescription. Then, use the Bayes theorem to find the probability that the patient followed a course of medication and yoga.

Complete step-by-step answer:
We shall let $A$ denote the event that a person has a heart attack. Let ${E_1}$ be the event that the person selected yoga and medication and ${E_2}$ be the event that the person adopted drug prescription.
We are given that chances of a patient having a heart attack is 40%.
In the fraction form, we get,
$P\left( A \right) = \dfrac{{40}}{{100}}$
Now, the person can choose any of the methods.
There are two methods, one method is yoga and medication and another method is the person adopting a drug prescription.
Then, $P\left( {{E_1}} \right) = P\left( {{E_2}} \right) = \dfrac{1}{2}$
We are given that the person who takes yoga and medication , the risk of heart attack is 70% of $P\left( A \right)$
Then, $P\left( {A|{E_1}} \right) = P\left( A \right)P\left( {{E_1}} \right) = \dfrac{{40}}{{100}}\left( {\dfrac{{70}}{{100}}} \right) = \dfrac{{28}}{{100}}$
Similarly, after adopting the drug prescription, the risk of heart rate is 75% of $P\left( A \right)$.
Hence, $P\left( {A|{E_2}} \right) = P\left( A \right)P\left( {{E_2}} \right) = \dfrac{{40}}{{100}}\left( {\dfrac{{75}}{{100}}} \right) = \dfrac{{30}}{{100}}$
But, we have to find the probability that the patient followed a course of medication and yoga.
We will use Bayes Theorem to find $P\left( {{E_1}|A} \right)$
Then, $P\left( {{E_1}|A} \right) = \dfrac{{P\left( {{E_1}} \right)P\left( {A|{E_1}} \right)}}{{P\left( {{E_1}} \right)P\left( {A|{E_1}} \right) + P\left( {{E_2}} \right)P\left( {A|{E_2}} \right)}}$
We will substitute the values in the above equation.
$P\left( {{E_1}|A} \right) = \dfrac{{\dfrac{1}{2}\left( {\dfrac{{28}}{{100}}} \right)}}{{\left( {\dfrac{1}{2}} \right)\left( {\dfrac{{28}}{{100}}} \right) + \left( {\dfrac{{30}}{{100}}} \right)\left( {\dfrac{1}{2}} \right)}} = \dfrac{{28}}{{58}} = \dfrac{{14}}{{29}}$
Hence, the probability that the patient followed a course of medication and yoga is $\dfrac{{14}}{{29}}$.

Note: Bayes theorem helps us to find the probability of an event when certain probability of events is given. Students must know how to apply Bayes theorem. The probability of the event is given as $\dfrac{{{\text{number of favourable outcome}}}}{{{\text{number of total outcomes}}}}$. When the probability of any event is given as a percentage, then the total outcome is considered as 100.