Probability with an unknown card You have a deck with 10 red cards and 10 black cards. Then you add one more card -- unknown whether red or black.
The 21 cards are shuffled. You pick randomly and pull a red card. How does that change the probability that the unknown card is a red card?
 A: Let us believe (for the duration of this answer) that probability does apply to this question.  Bayes' Rule gives us the optimal mechanism for updating our prior (to picking a card) beliefs about the probability that the added card was red based upon our observed draw of a red card.
We'll denote our prior probability that the card is red by $\pi$ and our posterior (after seeing the results of the card we picked) probability by $\pi'$.  The update from $\pi$ to $\pi'$ works as follows:
$$\pi' = {p(\text{draw red}|\text{red card})\cdot\pi \over p(\text{draw red}|\text{red card})\cdot\pi + p(\text{draw red}|\text{black card})\cdot(1-\pi)} $$
In this case, the probability that we draw a red card if the added card was red is $11/21$, and the probability that we draw a red card given that the added card was black is $10/21$.  Substituting gives:
$$\pi' = {11\cdot\pi \over 11\cdot\pi + 10\cdot(1-\pi)} $$
where we have eliminated the division by $21$ from the top and the bottom through the usual process of cancellation.  For concreteness, if we have a prior probability that the card is red of $\pi = 0.5$, the posterior probability that the card is red is:
$$\pi' = {11\cdot 0.5 \over 11\cdot 0.5 + 10\cdot 0.5} = 0.52381...$$
