2017 ACM-ICPC 亚洲区(西安赛区)网络赛 B Coin(逆元,费马小定理)
2017-09-30 18:27
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Bob has a not even coin, every time he tosses the coin, the probability that the coin's front face up is \frac{q}{p}(\frac{q}{p}
\le \frac{1}{2})pq(pq≤21).
The question is, when Bob tosses the coin kk times,
what's the probability that the frequency of the coin facing up is even number.
If the answer is \frac{X}{Y}YX,
because the answer could be extremely large, you only need to print (X
* Y^{-1}) \mod (10^9+7)(X∗Y−1)mod(109+7).
indicates the number of test cases (T
\le 100T≤100).
Then Each line has 33 integer p,q,k(1\le
p,q,k \le 10^7)p,q,k(1≤p,q,k≤107) indicates
the i-th test case.
2017
ACM-ICPC 亚洲区(西安赛区)网络赛
题意:抛k次硬币,求正面朝上为偶次数的概率,结果余1e9+7
思路:设向上概率概率是 a 向下为 b
k次扔取时,向上为偶数的为: C(k,0) *a^0 * b^k + C(k,2) *a^2 *b^(k-2)... + C(k,k)*a^k*b^0
所以结果应为: ((a+b)^k +(a-b)^k )/2
又a+b=1,a=p/q,所以应为(((p-2q)/p)^k+1)/2
关于除法逆元有(a / b) % p = (a * inv(a) ) % p = (a % p * inv(a) % p) % p
上代码:
\le \frac{1}{2})pq(pq≤21).
The question is, when Bob tosses the coin kk times,
what's the probability that the frequency of the coin facing up is even number.
If the answer is \frac{X}{Y}YX,
because the answer could be extremely large, you only need to print (X
* Y^{-1}) \mod (10^9+7)(X∗Y−1)mod(109+7).
Input Format
First line an integer TT,indicates the number of test cases (T
\le 100T≤100).
Then Each line has 33 integer p,q,k(1\le
p,q,k \le 10^7)p,q,k(1≤p,q,k≤107) indicates
the i-th test case.
Output Format
For each test case, print an integer in a single line indicates the answer.样例输入
2 2 1 1 3 1 2
样例输出
500000004 555555560
题目来源
2017ACM-ICPC 亚洲区(西安赛区)网络赛
题意:抛k次硬币,求正面朝上为偶次数的概率,结果余1e9+7
思路:设向上概率概率是 a 向下为 b
k次扔取时,向上为偶数的为: C(k,0) *a^0 * b^k + C(k,2) *a^2 *b^(k-2)... + C(k,k)*a^k*b^0
所以结果应为: ((a+b)^k +(a-b)^k )/2
又a+b=1,a=p/q,所以应为(((p-2q)/p)^k+1)/2
关于除法逆元有(a / b) % p = (a * inv(a) ) % p = (a % p * inv(a) % p) % p
上代码:
#include<iostream> a9d4 #include<cstdio> #include<cstring> #include<string> #include<cmath> #include<queue> #include<stack> #include<vector> #include<algorithm> using namespace std; const int maxn = 10000 + 10; const double pi = acos(-1); #define ll long long #define inf 0x3f3f3f3f #define llinf 0x3f3f3f3f3f3f3f3f #define mem(a,b) memset(a,b,sizeof(a)); const ll mod = 1e9 + 7; ll q_mod(ll a, ll b) { ll res = 1; while (b) { if (b & 1) res = res*a%mod; a = a*a%mod; b >>= 1; } return res; } int main() { //freopen("Text.txt", "r", stdin); int t; ll p, q, k; scanf("%d", &t); while (t--) { scanf("%lld%lld%lld", &p, &q, &k); ll ans = (p - 2 * q)*q_mod(p, mod - 2) % mod;//除法变乘法 ans = q_mod(ans, k); ans = (ans + 1) % mod; ans = ans*q_mod(2, mod - 2) % mod; printf("%lld\n", ans); } return 0; }
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