Question 1163319
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You are given two urns, each containing a collection of coloured balls. 
Urn 1 contains three white and two blue balls. 
Urn 2 contains four white and three blue balls. 
A ball is drawn at random from urn 1 and put into urn 2. 
A ball is then picked a random from Urn 2.
What is the probability that the ball picked is white? Assume balls picked from an urn are equally likely to be any of the balls present.
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<pre>
(1)  Analyzing the first step, when we draw the ball from Urn 1 and put it into Urn 2:


         After the first step, we draw the white ball from Urn 1 with the probability of  {{{3/(3+2)}}} = {{{3/5}}} and put it into Urn 2.

         So, with the probability  {{{3/5}}}  we get the content (5white,3blue) in the Urn 2 at this step.


         Alternatively, with the complement probability  {{{2/5}}}  we get the content (4white,4blue) in Urn 2 at this step.




(2)  Analyzing the second step, when we draw a ball from Urn 2, we have two respective alternatives


         we draw a white ball from Urn 2  (5w,3b)  with the probability  {{{5/(5+3)}}} = {{{5/8}}},  or

         we draw a white ball from Urn 2  (4w,4b)  with the probability  {{{4/(4+4)}}} = {{{4/8}}}.




(3)  Now the probability to have finally a white ball from Urn 2 is


         P = {{{(3/5)*(5/8)}}} + {{{(2/5)*(4/8)}}} = {{{15/40}}} + {{{8/40}}} = {{{23/40}}}.    <U>ANSWER</U>
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Solved.