Sharad Talvalkar

Explain prior probability & posterior Probability along with application of   Bayes Theorem in a business scenario.
 
Prior probability : Probability is an intricate subject. Therefore, initially the concepts of Probability  are introduced / explained  with the  help of orderly examples where the outcomes are known to us by applying simple logic/ common sense. E.g. Tossing a fair coin, rolling an unbiased dice, drawing a card from a deck. In  all these aforesaid cases we can make our probability statements even prior to conducting any experiment.  We know that the probability of getting a Head when a coin is tossed is 0.5 ( 50%) Hence such classical cases are known as Prior Probabilities where the outcome is known even before the experiment is conducted.
Classical approach defines the probability of getting either Head or Tail when a coin is tossed as

This approach to probability is useful when we deal with Coin tosses, card games , dice game etc. Real life situations in management are not  so straight forward & therefore one has to define probability in a different way.
Posterior Probability :     At the beginning of the World Cup Cricket Match 2019, Indian Fans were very confident (99%)  that India will win the world cup. As the matches progressed, some key players like Shikhar Dhawan , Vijay Shankar got  injured & they could not participate .  Hence after getting this additional information, Indian Fans revised the probability  ( let’s say 80% )of winning a world cup. This revised probability after getting additional information is known as Posterior Probability.  
A similar situation also occurs in a business scenario. A shopkeeper may order various colors of Jacket ( Blue, Black, Grey etc.) based on the past consumption pattern. As time progresses, he may notice that the sale of Jackets is not as per his expectation & so after getting this input the shopkeeper  may change his ordering pattern of Jackets. This revised ordering pattern is an example of Posterior Probability.
 
Bayes Theorem: Bayes formula for conditional probability under dependence is as follows

Let us now understand the application of Bayes Theorem in a business scenario with the help of following example
Suppose there are three machines ( M1,M2 & M3), each of them producing a same component, say X. Production from M1, M2 & M3 is 40%, 49% & 11%. If there is a customer complaint from the market what is the probability that it is from M1, M2 & M3.
 Using simple logic, we can say that the probability of defective coming from M1,M2 & M3 is 0.4, 0.49 & 0.11 respectively.
Now we have additional information that the defectives from M1,M2 & M3 are 0.5%, 3% & 2%. In this scenario when a complaint  comes from the market what is the probability that the defective is coming from M1,M2 & M3 .
Using Bayes Theorem, we can say that

 
Using Bayes Theorem, we can say that
In the above example our initial probability of getting defectives from M1,M2 & M3 was 0.4, 0.49 & 0.11.respectively. This probability is the Prior Probability.
 
 
Later, after getting additional information that the defectives from M1,M2 & M3 are 0.5%, 3% & 2% we have revised the probability of getting defectives from M1,M2, M3 to 0.1058, 0.7778 & 0.1164. This revised probability is known as Posterior Probability.    
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Bayes Theorem.docx