An urn contains $2$ white and $2$ black balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with another ball of same colour. The process is repeated. The probability that the third ball drawn is black is $1 - \frac{k}{30}$ .Find the value of $k$ ?

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Solution

Let $B_{i} =$ ith ball drawn is black
$W_{i} =$ it ball drawn is white; $i = 12$
and $A =$ third ball drawn is black.
We observe that the black ball can be drawn in the third draw in one of the following mutually exclusive ways.
1) Both first and second balls drawn are white and third ball drawn is black,
i.e $(W_{1} \cap W_{2}) \cap A$
2) Both first and second balls are black and third ball drawn is black
i.e. $(B_{1} \cap B_{2}) \cap A$
3) The first ball drawn is white, the second ball drawn is black and the third ball drawn is black
i.e $(W_{1} \cap B_{2}) \cap A$
4) The first ball drawn is black, the second ball drawn is white and the third ball drawn is black
i.e $v$
$∴ P (A) = p [{(W_{1} \cap W_{2}) \cap A} \cup {(B_{2} \cap B_{2}) \cap A} \cup {(W_{1} \cap B_{2}) \cap A} \cup {(B_{1} \cap W_{2}) \cup A}] = P {(W_{1} \cap W_{2}) \cap A} + P {(B_{1} \cap B_{2}) \cap A} + P {(W_{1} \cap B_{2}) \cap A} + P {(B_{1} \cap W_{2}) \cap A}$
$= P (W_{1} \cap W_{2}) . P (\frac{A}{(W_{1} \cap W_{2})}) + P (B_{1} \cap B_{2}) . P (\frac{A}{(B_{1} \cap B_{2})})$
$+ P (W_{1} \cap B_{2}) . P (\frac{A}{(W_{1} \cap B_{2})}) + P (B_{1} \cap W_{2}) . P (\frac{A}{(B_{1} \cap W_{2})})$
$= (\frac{2}{4} \times \frac{1}{3}) \times 1 + (\frac{2}{4} \times \frac{3}{5}) \times \frac{4}{6} + (\frac{2}{4} \times \frac{2}{3}) \times \frac{3}{4} + (\frac{2}{4} \times \frac{2}{5}) \times \frac{3}{4}$
$= \frac{1}{6} + \frac{1}{5} + \frac{1}{4} + \frac{3}{20} = \frac{23}{30}$

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An urn contains 2 white and 2 black balls. A ball is drawn at random. If it is white it is not replaced into the urn. Otherwise it is replaced along with another ball of same colour. The process is repeated. The probability that the third ball drawn is black is1−k30.Find the value of k ?