tailieunhanh - Báo cáo toán học: "A simple card guessing game revisited"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A simple card guessing game revisited. | A simple card guessing game revisited Arnold Knopfmagher The John Knopfmacher Centre for Applicable Analysis and Number Theory Department of Applied Mathematics University of the Witwatersrand P. O. Wits 2050 Johannesburg South Africa email arnoldk@ homepage http science number_theory Helmut Prodinger The John Knopfmacher Centre for Applicable Analysis and Number Theory Department of Mathematics University of the Witwatersrand P. O. Wits 2050 Johannesburg South Africa email helmut@ homepage http helmut Submitted January 4 2000 Accepted March 3 2000. Abstract A deck of cards consisting of m red and n black cards is given. A guess is made as to the colour of the top card after which it is revealed and discarded. To maximise the number of correct guesses one chooses the colour corresponding to the majority of cards remaining in the deck. We determine the expected number of correct guesses with this strategy as well as the probability of beating an opponent who uses the naive stategy of random guessing. AMS Subject Classification. 05A15 primary 60C05 secondary the electronic journal of combinatorics 8 no. 2 2001 R13 1 1 Introduction A number of authors have considered the following simple combinatorial game 5 3 7 6 . One starts with a deck of cards consisting of m red and n black cards. A guess is made as to the colour of the top card after which it is revealed and discarded. To maximise the number of correct guesses one chooses the colour corresponding to the majority of cards remaining in the deck. We determine the expected number of correct guesses with this strategy as well as the probability of beating an opponent1 who uses the naive stategy of random guessing. The hrst of these problems was analysed by Levasseur 3 but only for the case m n. The second question was investigated by Zagier 7 again for the case m n. He obtained an asymptotic answer only whereas we can derive .