PS8-5HW We play a game with a deck of 52 regular playing cards, of which 26 are

Question

PS8-5HW We play a game with a deck of 52 regular playing cards, of which 26 are red and 26 are black. I randomly shuffle the cards and place the deck face down on a table. You have the option of "taking" or "skipping" the top card. If you skip the top card, then that card is revealed and we continue playing with the remaining deck. If you take the top card, then the game ends. If we get to a point where there is only one card left in the deck, you must take it. You win if the card you took was revealed to be black, and you lose if it was red. Prove that you have no better strategy than to take the top card-which means your probability of winning is 1/2. 8 points] Hint: Prove by induction the more general claim that for a randomly shuffled deck of n cards that are red or black- not necessarily with the same number of red cards and black cards-there is no better strategy than taking the top card. To precisely state this more general claim, first work out your probability of winning if you simply take the top card

Explanation / Answer

Probability of drawing a random card to turn out black = 26/52=1/2

Lets take the first card which is revealed to be red. P( black ) in 2nd round = 26/51

If the first car was black, P( black ) in 2nd round = 25/51

P( black in 2nd round) = 1/2[25+26]/51 = 1/2

Thus, it can be seen that probbility remains 1/2 only

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