#include <iostream>
#include <vector>
#include <iomanip>

const double alphpa = 0.5; //Probability to miss the snowball
const size_t N = 10000; //Amount of terms in the sum

int main()
{
    std::cout << std::setprecision(30);
    double complementary_alpha = 1.0 - alphpa;
    std::vector<double> a, b, result;
    a.push_back(1.0);
    a.push_back(0.0);
    b.push_back(a[0]);
    b.push_back(alphpa * b[0] + a[1]);
    result.push_back(b.back());
    //Print probability of getting 0 points
    //std::cout << 0 << '\t' << result.back() << std::endl;
    for (size_t i = 1; i <= N; ++i) {
        a.push_back(0.0);
        b.push_back(0.0);
        for (size_t j = 0; j <= i - 1; ++j) {
            a[j] = complementary_alpha * b[j];
        }
        a[i] = 0.0;
        b[0] = a[0];
        for (size_t j = 1; j <= i + 1; ++j) {
            b[j] = alphpa * b[j - 1] + a[j];
        }
        result.push_back(b[i + 1]);
        //Print probability of getting 10i points
        //std::cout << 10 * i << '\t' << result.back() << std::endl;
    }

    ////Print partial sums of probabilities of getting each amount of points
    //for (size_t i = 0; i < result.size(); ++i) {
    //    double sum = 0.0;
    //    for (size_t j = 0; j <= i; ++j) {
    //        sum += result[j];
    //    }
    //    std::cout << sum << std::endl;
    //}

    double expectation = 0.0;
    for (size_t i = 0; i < result.size(); ++i) {
        expectation += 10 * i * result[i];
        //Print partial sums of average amount of points
        //std::cout << expectation << std::endl;
    }
    //Print average amount of points
    std::cout << expectation << std::endl;

    return 0;
}