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Understanding Bayes’ Theorem


Therefore the probability of the car being actually blue, when the witness identified it as blue equals: (0.56/0.62) 0.903

 

Question: Sum Rule & Bayes’ Theorem

Jack has 3 white balls and 2 red balls in his box while his friend Andrew has 4 white and 5 red balls
Andrew took 1 white ball from jack and gave him 1 red ball in compensation. Now calculate the probability of picking a red ball from Andrew’s box
After the exchange, Tom stole a ball from one of the boxes and found that it’s white. If you have to tell who lost his white ball, what should be your say?  
 

Solution

 

 

 

 

Solution

  • Initially Jack has 3 white balls and 2 red balls in his box while his friend Andrew has 4 white and 5 red balls
  • After the exchange Jack has 2 white and 3 red balls, and Andrew has 5 white and 4 red balls. Therefore the probability of picking a red ball from Andrew’s box is:
  • P(RAndrew) = 4 / (5+4) = 4/9
  • Now Tom stole a white ball from one of the two boxes. To make a calculated guess about who lost 1 white ball, we need to calculate the conditional probabilities.
  • P(Jack’s box/If the balls is White)= Probability of white balls in Jack’s box/(Probability of white ball in Jack’s box +Probability of white ball in Andrew’s box)
  • Similarly, P(Andrew’s box /White)  
  • Point to note here is that the white ball can come from 2 boxes only, so the sum of conditional probabilities of the boxes, given the ball is white should sum to 1, which is (18/43+25/43) = 1 in our case.





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