We play a game with a deck of 52 regular playing cards, of which 26 arered...

We play a game with a deck of 52 regular playing cards, of which 26 arered and 26 are black. They’re randomly shuffled and placed face down ona table. You have the option of “taking” or “skipping” the top card. Ifyou skip the top card, then that card is revealed and we continue playingwith the remaining deck. If you take the top card, then the game ends;you win if the card you took was revealed to be black, and you lose if itwas red. If we get to a point where there is only one card left in the deck,you must take it. Prove that you have no better strategy than to take thetop card – which means your probability of winning is 1/2.

Hint: Prove by induction the more general claim that for a randomlyshuffled deck ofncards that are red or black – not necessarily with thesame number of red cards and black cards – there is no better strategy than taking the top card

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