14. A ball was added to an urn containing one white ball - either white or black (with equal probabilities of selection). After this, one ball was randomly drawn from the urn. It turned out to be white. What is the conditional probability that the remaining ball in the urn is also white?
Solution. Let

$$
\begin{aligned}
A & =\{\text { a white ball was added to the urn }\}, \\
C & =\{\text { a white ball was drawn }\}, \\
D & =\{\text { a white ball remains }\} .
\end{aligned}
$$

Note that $A=C D$. Then

$$
\begin{aligned}
\mathrm{P}(D \mid C)=\frac{\mathrm{P}(D C)}{\mathrm{P}(C)}=\frac{\mathrm{P}(A)}{\mathrm{P}(C \mid A) \mathrm{P}(A)+\mathrm{P}(C \mid \bar{A}) \mathrm{P}(\bar{A})} & \\
& =\frac{1 / 2}{1 \cdot 1 / 2+1 / 2 \cdot 1 / 2}=\frac{2}{3} .
\end{aligned}
$$