Problem 2. [20 pts I have three urns, Ui, U2 and Us, such the first urn contains

Question

Problem 2. [20 pts I have three urns, Ui, U2 and Us, such the first urn contains w identical white balls and b identical black balls, the second urn contains w identical white balls and b identical black balls, and the third urn contains ws identical white balls and ba identical black balls. I first choose a ball uniformly at random from Ui (i.e., every ball is equally likely to be picked) and drop it into U. Next, I pick a ball uniformly at random fron U2 and drop it into U. Finally, I pick a ball uniformly at random from Us and it turns out to be white. What is the probability that the ball that was transferred from U to U2 was black? Show the main steps of your solution Hint: Note that this problem is asking for a conditional probability. Use the tree method

Explanation / Answer

The contents of three urns 1,2 and 3 are as follows

urn 1 w1 white and b1 black balls

urn 2 w2 white and b2 black balls

urn 3 w3 white and b3 black balls

let us suppose that the ball transpormed from urn 1 to urn 2 is is black then the number of black balls in the urn 3 becomes b4 then one ball selected from urn 3 and is to be white its probability is given below

step 1 probability of selecting one black ball from urn 1 is equal to b1c1 / (w1 + b1)

step 2 a probability of selecting i white ball from urn 2 after entering 1 black ball from urn 1 is given as

  W2 C1 / (w2 +b2) +1

step 2b. probability of selecting 1 black ball from urn 2 after entering 1 black ball from bag 1 is given as (b2 +1)C 1 / (w2+b2)+1

step 3a. probability of selecting one white ball from urn 3 after entering one white ball from urn 2 is given as

(w3+1)C 1 / (w3+b3)+1

step 3b. probability of selecting one white ball after entering one black ball from urn 2 is given as

w3 C1 / (w3+b3)+1

Now the required probability of getting one white ball from urn 3 is given as (step 1) x(step 2a)x(step3a) +(step 1) x(step 2b)x(step3a) + (step 1) x(step 2a)x(step3b)+(step 1) x(step 2b)x(step3b)this is the probabilitynof getting one white ball from urn 3 after entering one black ball to urn 2 from urn 1

  

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