A standard 52-card deck has 26 red and 26 black cards and is uniformly shuffled. Discard the first 10 cards without observing their identities. What is the probability that the next card is red? Give two derivations: one using exchangeability of positions and one conditioning on the number of red cards discarded. Explain how the answer changes if the colors of the discarded cards are observed. ### Constraints & Assumptions - The shuffle is uniform. - Discarded card identities and colors are not observed for the main question. - The eleventh card is drawn from the same shuffled deck without reshuffling. ```hint Every position has the same marginal distribution Before any card information is observed, position eleven is as likely to hold each original card as position one. ``` ### What a Strong Answer Covers - A direct symmetry argument. - A conditional calculation over the number of red discards and the law of total expectation. - The difference between unobserved discards and conditioning on observed colors. ### Follow-up Questions - If exactly six discarded cards were red, what is the new probability? - Does the number of discarded cards matter when their colors are hidden? - Are the colors of positions eleven and twelve independent?