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UESTC 890 Card Trick(DP 纸牌魔术)

2024-07-19 23:04:52   220人阅读

题意  给你一些牌  全部正面朝下放桌子上   你选一个起点    翻开那张牌   牌上的数字是几就向前走几步   J,Q,K 都是向前走10步  A向前走11步   知道向前走对应的步数后超过了终点   输入n m 和n个数   代表你以第m张牌为起点   依次掀开了n张牌就不能再掀了    然后同样的牌   Alice以1-10张牌中的任意一个为起点   求Alice最后的终点与你的终点相同的概率


c[i]表示第i张牌的面值   没被掀开的牌的面值都是未知的c[i]=0  可能为2-A中的任意一个  令d[i]表示从你的终点到达第i张牌的概率   那么所有掀开过的牌的概率都为1   然后从终点向前递推   当p[i]=0时   p[i]=sum{p[i+j]}  j为2-A中任意一张牌  注意10,j,q,k的时候都是10   最后的答案就是1到10的结果加起来除以10了



#include<cstdio>
#include<cstring>
using namespace std;
const int N = 1500;

int main()
{
    char s[3];
    int n, m, l;
    double p[N], ans;
    while (~scanf ("%d%d", &n, &m))
    {
        memset (p, 0, sizeof (p));
        l = m;

        for (int i = 1; i <= n; ++i)
        {
            scanf ("%s", s);
            p[l] = 1;
            if (s[0]<'A' && s[1]!='0')  l += s[0] - '0';
            else if (s[0] == 'A')  l += 11;
            else l+= 10;
        }

        ans = 0;
        for (int i = l ; i >= 1; --i)
        {
            if (p[i] == 0)
            {
                for (int j = 2; j <= 11; ++j)
                {
                    int t = (j == 10 ? 4 : 1);
                    p[i] += t * p[i + j];
                }
                p[i] /= 13;
            }
            if (i <= 10) ans += p[i];
        }

        printf ("%.8f\n", ans / 10);
    }
    return 0;
}

Card Trick

Time Limit: 2999/999MS (Java/Others)     Memory Limit: 65432/65432KB (Java/Others)
 

I am learning magic tricks to impress my girlfriend Alice. My latest trick is a probabilistic one, i.e. it does work in most cases, but not in every case. To perform the trick, I first shuffle a set of many playing cards and put them all in one line with faces up on the table. Then Alice secretly selects one of the first ten cards (i.e. she chooses x0, a secret number between 1 and 10 inclusive) and skips cards repeatedly as follows: after having selected a card at position xi with a number c(xi) on its face, she will select the card at position xi+1=xi+c(xi). Jack (J), Queen (Q), and King (K) count as 10, Ace (A) counts as 11. You may assume that there are at least ten cards on the table.

Alice stops this procedure as soon as there is no card at position xi+c(xi). I then perform the same procedure from a randomly selected starting position that may be different from the position selected by Alice. It turns out that often, I end up at the same position. Alice is very impressed by this trick.

However, I am more interested in the underlying math. Given my randomly selected starting position and the card faces of every selected card (including my final one), can you compute the probability that Alice chose a starting position ending up on the same final card? You may assume that her starting position is randomly chosen with uniform probability (between 1 and 10 inclusive). I forgot to note the cards that I skipped, so these cards are unknown. You may assume that the card face of every single of the unknown cards is independent of the other card faces and random with uniform probability out of the possible card faces (i.e. 2-10, J, Q, K, and A).

title

Illustration of first sample input: my starting position is 2, so I start selecting that card. Then I keep skipping cards depending on the card‘s face. This process iterates until there are not enough cards to skip (in this sample: Q). The final Q card is followed by 0 to 9 unknown cards, since Q counts as 10.

Input

For each test case:

  • A line containing two integers n (1≤n≤100) and m (1≤m≤10) where n is the number of selected cards and m is the 1-based position of my first selected card.
  • A line with n tokens that specify the n selected card faces (in order, including the final card). Each card face is given either as an integer x (2≤x≤10) or as a single character (J, Q, K, or A as specified above).

Output

For each test case, print one line containing the probability that Alice chooses a starting position that leads to the same final card. Your output should have an absolute error of at most 10?7.

Sample input and output

Sample InputSample Output
5 2
2 3 5 3 Q
1 1
A
1 2
A
1 10
A
6 1
2 2 2 2 2 2
7 1
2 2 2 2 2 2 2
3 10
10 J K
0.4871377757023325348071573
0.1000000000000000000000000
0.1000000000000000000000000
0.1748923357025314239697490
0.5830713210321767445117468
0.6279229611115749556280350
0.3346565827603272001891974





UESTC 890 Card Trick(DP 纸牌魔术)


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