摘要
一副牌有紅色及黑色各若干張，充分洗勻後，玩家猜牌組中最上面那張的顏色，猜完後公開該張的顏色且不放回，持續進行這樣的猜牌，直到整副牌猜完才停止。若想使得猜中張數的期望值最大，顯然應該每次都猜所剩牌組中佔多數的顏色；文獻上，Knopfmacher和Prodinger(2001)與Sulanke(1995)分別給出上述猜牌策略下猜中張數的期望值。針對這類問題，我們在原本的遊戲規則中加入求救機制「SOS」：玩家擁有一次機會可以在遊戲進行至任何一張牌時，要求莊家翻開該張牌，且這張算在玩家猜中張數裡；我們給出此時玩家猜牌的最佳策略及猜中張數的期望值。更進一步探討，當玩家可使用求救次數是t次時，玩家猜牌的最佳策略及猜中張數的期望值。最後我們限制玩家猜紅及猜黑的各別次數；在此限制下，我們可以證明：無論玩家如何猜牌，其猜中張數的期望值皆相同。
並列摘要
A deck of cards consisting of m red and n black cards is given. A guess is made as to the color of the top card, after which it is revealed and discarded. Continue guessing until the deck is exhausted. To maximize the number of correct guesses one chooses the color corresponding to the majority of cards remaining in the deck. Recently, Knopf-macher and Prodinger (2001) and Sulanke (1995) determined the expected number of correct guesses with this strategy by using different techniques. In this paper, we pro-pose two variants of this game. In the first variant, the player can use one time of SOS at any card. After using an SOS at some card, this card will be revealed and discarded, and the total number of correct guesses increases by one. Moreover, the guessing game in which the player can use t times of SOS will also be investigated. For each of these two guessing games, we will find the optimal strategy and the corresponding expected number of correct guesses. Finally, we study another variant that restricts the number of guesses of each color. In the restricted guessing game, we prove that the expected number of correct guesses does not depend on the guessing strategy.
參考文獻
延伸閱讀
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