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Two urns contain, respectively, $20$ white and $10$ black balls and $10$ white and $50$ black balls. $5$ balls are transferred from the first urn to the second urn, and then $1$ ball is drawn from the second urn and is found to be white. What is the probability that this white ball comes form first urn?

I can find the probability of 'the ball drawn from the 2nd urn is found to be white', I also can find the probability of 'the five balls transferred are all white when it is known that we found the ball from the 2nd urn is white'... but this particular Question is very confusing

How do I find the probability of drawing a white ball from the second urn?

let

$W =$ White, $B= $ Black

$X \equiv$ Event , founding a white ball from 2nd urn.

Following events are considered when transferring the balls from 1st urn to 2nd urn :

$A \equiv $ Event, $(1W,4B)$

$B \equiv $ Event, $(2W,3B)$

$C \equiv $ Event, $(3W,2B)$

$D \equiv $ Event, $(4W,1B)$

$E \equiv $ Event, $(5W,0B)$

$P(X) = P(X|A)P(A)+ P(X|B)P(B)+P(X|C)P(C)+P(X|D)P(D)+P(X|E)P(E)$

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  • $\begingroup$ What do you mean by : "and then 1 ball is drawn form the second urn and is found to be white" $\endgroup$ – Surb Sep 28 '15 at 10:27
  • $\begingroup$ after transferring 5 balls arbitrarily from 1st urn to 2nd urn, one ball is drawn from the 2nd urn and found to be white.. $\endgroup$ – DEEP Sep 28 '15 at 10:31
  • $\begingroup$ When you pose a question here, it is expected that you include your own attempt to solve the problem and indicate where you are stuck so that you receive responses that address the specific difficulties you are encountering. $\endgroup$ – N. F. Taussig Sep 28 '15 at 10:34
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    $\begingroup$ Consider cases: When the five balls are transferred from the first urn to the second, the number of white balls that are transferred can vary from $0$ to $5$. $\endgroup$ – N. F. Taussig Sep 28 '15 at 11:17
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    $\begingroup$ Please do not use all-caps in your question. This is basically the most universal rule on the internet so you should've figured it's not appropriate on maths.SE either. $\endgroup$ – Lord_Farin Sep 28 '15 at 15:04
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A good first step in trying to apply Bayes' theorem is to write it down. There are a few different ways people do this, but the formula could look like this: $$ P(B \mid A) = \frac{P(A \cap B)}{P(A)} = \frac{P(A \mid B)\,P(B)}{P(A)}. $$

In order to apply the theorem, you need $A$ and $B$ in the formula to match two events you can describe in your problem. Since you want to know the probability that the ball drawn from the second urn came originally from the first urn, given that the ball drawn from the second urn is white, and since $P(B \mid A)$ can be read "probability of $B$ given $A$," the obvious way to apply the theorem is to set \begin{align} A &= \text{the ball drawn from the second urn is white}, \\ B &= \text{the ball drawn from the second urn came originally from the first urn}. \end{align}

You say you found $P(A)$ already, so you next might want to try to find $P(A\cap B)$ or to find $P(B)$ and $P(A \mid B)$.

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  • $\begingroup$ Yes, i also do in this way ... but event $A\cap B$ and event $B$ is not cleared to me $\endgroup$ – DEEP Sep 28 '15 at 11:21
  • $\begingroup$ Of all the probabilities you might need, $P(B)$ should be the easiest to find: just before you draw the ball from the second urn, how many balls were in that urn and how many of those came from the first urn? Then you only need $P(A\mid B)$, which also is not hard if you think about what it really means. $\endgroup$ – David K Sep 28 '15 at 11:36
  • $\begingroup$ okay before drawing the ball from the 2nd urn there are total 65 balls in the 2nd urn. and 5 of them are from 1st urn $\endgroup$ – DEEP Sep 28 '15 at 11:42
  • $\begingroup$ i think $P(B)= {5 \over 65}$ ... isn't it?? $\endgroup$ – DEEP Sep 28 '15 at 11:47
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    $\begingroup$ I get the same answer in the same way. $\endgroup$ – David K Sep 28 '15 at 11:53
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Hint

Say that the white balls of the first box are in fact blue. So, what is the probability to draw on a blue ball ?

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  • $\begingroup$ probability of finding blue ball is 20 /30 $\endgroup$ – DEEP Sep 28 '15 at 10:53
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    $\begingroup$ @DEEP If you only draw one ball from the first urn, probability that that ball is blue is then $20/30$. But that tells you little about drawing five balls from the first urn or one ball from the second urn. $\endgroup$ – David K Sep 28 '15 at 10:58
  • $\begingroup$ still confused.. :-( $\endgroup$ – DEEP Sep 28 '15 at 11:07

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