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# 面试题60

把n个骰子扔在地上，所有骰子朝上一面的点数之和为s。输入n，打印出s的所有可能的值出现的概率。

你需要用一个浮点数数组返回答案，其中第 i 个元素代表这 n 个骰子所能掷出的点数集合中第 i 小的那个的概率。

```
示例 1:

输入: 1
输出: [0.16667,0.16667,0.16667,0.16667,0.16667,0.16667]

示例 2:

输入: 2
输出: [0.02778,0.05556,0.08333,0.11111,0.13889,0.16667,0.13889,0.11111,0.08333,0.05556,0.02778]
```

## 限制：

* 1 <= n <= 11

## Solutions

1. **dynamic programming**
2. Count the number of cases with sum equals to values(sums) ranging from `n` to `n * 6` after rolled `n` times.
3. Then the propability of each `value(sum)` is `count * pow(1.0/6, n)`. ie: totoal number of combinations is `6^n`
4. `dp[n][i]` represents the number of cases with sum equals to `i` after we rolled `n` times.
   * Then `dp[n + 1][i] = dp[n][i - 6](current dice is 6) + dp[n][i - 5](current dice is 5) + .... dp[n][i - 1]`.

```cpp
class Solution {
public:
    vector<double> twoSum(int n) {
        vector<int> dp(n * 6 + 1, 0);
        for (int i = 1; i <= 6; i++)
            dp[i] = 1;

        for (int i = 2; i <= n; i++) {
            // traverse backwards to avoid overwrite data in dp table of the last time
            for (int sum = i * 6; sum >= i; sum--) {
                // Caution, must use a temporal varaible
                int cnt = 0;
                for (int cur = 1; cur <= 6; cur++) {
                    int prev = sum - cur;
                    if (prev < i - 1 || prev > (i - 1) * 6)
                        continue;
                    cnt += dp[prev];
                }
                dp[sum] = cnt;
            }
        }

        vector<double> res;
        double base = pow(1.0 / 6, n);
        for (int sum = n; sum <= n * 6; sum++)
            res.push_back(dp[sum] * base);

        return res;
    }
};
```