Consider a deck of 52 cards. We shuffle the cards randomly; all possible permuta

Question

Consider a deck of 52 cards. We shuffle the cards randomly; all possible permutations are equally likely. We now form a hand of 26 cards as follows. We look at the cards two at a time; thus we look at cards 1 and 2 together, then cards 3 and 4 together, and so on until we look at cards 51 and 52. If we see only one heart we take the heart; if we see two hearts or no hearts it does not matter what we do, so for definiteness take the first card. This gives us a hand with 26 cards. What is the probability we have all 13 hearts in our 26 cards? Give an exact answer, as well as an approximation to four decimal digits. Write a computer program to simulate 1,000,000 deals and estimate the probability.

Explanation / Answer

typedef struct struct_card {
unsigned short int suit:2;
unsigned short int card:4;
// unsigned short int valu:4;
} card;

int main() {
card a_card;
card std_deck[52];
const unsigned short int rummy_value[13] = {1,2,3,4,5,6,7,8,9,10,10,10,10};
const char *std_card_name[13] = {"Ace","Two","Three","Four","Five","Six",
"Seven","Eight","Nine","Ten","Jack","Queen","King"};
const char *std_suit_name[4] = {"Spades","Clubs","Hearts","Diamonds"};

int j, k, i=0;
for(j=0; j<4; j++){
for(k=0; k<13; k++){
a_card.suit=j; a_card.card=k;
std_deck[i++] = a_card;
}
}

//check our work
printf("In a game of rummy: ");
for(i=0;i<52;i++){
printf(" %-5s of %-8s is worth %2d points. ",
std_card_name[std_deck[i].card],
std_suit_name[std_deck[i].suit],
rummy_value[std_deck[i].card]);
}

//a different kind of game.
enum round_mode {SHEILD_TRUMP, FLOWER_TRUMP, BELL_TRUMP, ACORN_TRUMP, BOCK, GEISS} mode;
const card jass_deck[36]={
{0,0},{0,1},{0,2},{0,3},{0,4},{0,5},{0,6},{0,7},{0,8},
{1,1},{1,1},{1,2},{1,3},{1,4},{1,5},{1,6},{1,7},{1,8},
{2,2},{2,1},{2,2},{2,3},{2,4},{2,5},{2,6},{2,7},{2,8},
{3,3},{3,1},{3,2},{3,3},{3,4},{3,5},{3,6},{3,7},{3,8},
};
#define JASS_V {11,0,0,0,0,10,2,3,4}
const unsigned short int jass_value[9] = JASS_V;
#define JASS_TRUMP_V {11,0,0,0,14,10,20,3,4}
const unsigned short int jass_trump_value[9] = JASS_TRUMP_V;
#define JASS_BOCK_V {11,0,0,8,0,10,2,3,4}
const unsigned short int jass_bock_value[9] = JASS_BOCK_V;
#define JASS_GEISS_V {0,11,0,8,0,10,2,3,4}
const unsigned short int jass_geiss_value[9] = JASS_GEISS_V;
const char *jass_card_name[9] = {"Ace","Six","Seven","Eight","Nine","Banner",
"Under","Ober","King"};
const char *jass_suit_name[4] = {"Sheilds","Flowers","Bells","Acorns"};
const unsigned short int jass_all_value[6][4][9] = {
{ JASS_TRUMP_V, JASS_V, JASS_V, JASS_V },
{ JASS_V, JASS_TRUMP_V, JASS_V, JASS_V },
{ JASS_V, JASS_V, JASS_TRUMP_V, JASS_V },
{ JASS_V, JASS_V, JASS_V, JASS_TRUMP_V },
{ JASS_BOCK_V, JASS_BOCK_V, JASS_BOCK_V, JASS_BOCK_V },
{ JASS_GEISS_V, JASS_GEISS_V, JASS_GEISS_V, JASS_GEISS_V }
};

//check our work 2: work goes on summer vacation
printf("In a game of jass with trump (Sheilds | Flowers | Bells | Acorns) | Bock | Geiss ");
for(i=0;i<36;i++){
printf(" %-6s of %-7s is worth %8d%10d%8d%9d%8d%8d ",
jass_card_name[jass_deck[i].card],
jass_suit_name[jass_deck[i].suit],
jass_all_value[SHEILD_TRUMP][jass_deck[i].suit][jass_deck[i].card],
jass_all_value[FLOWER_TRUMP][jass_deck[i].suit][jass_deck[i].card],
jass_all_value[BELL_TRUMP][jass_deck[i].suit][jass_deck[i].card],
jass_all_value[ACORN_TRUMP][jass_deck[i].suit][jass_deck[i].card],
jass_all_value[BOCK][jass_deck[i].suit][jass_deck[i].card],
jass_all_value[GEISS][jass_deck[i].suit][jass_deck[i].card]);
}
return 0;
}

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