<< problem 429 - Sum of squares of unitary divisors Powers With Trailing Digits - problem 455 >>

# Problem 436: Unfair wager

Julie proposes the following wager to her sister Louise.
She suggests they play a game of chance to determine who will wash the dishes.
For this game, they shall use a generator of independent random numbers uniformly distributed between 0 and 1.
The game starts with S = 0.
The first player, Louise, adds to S different random numbers from the generator until S > 1 and records her last random number x.
The second player, Julie, continues adding to S different random numbers from the generator until S > 2 and records her last random number y.
The player with the highest number wins and the loser washes the dishes, i.e. if y > x the second player wins.

For example, if the first player draws 0.62 and 0.44, the first player turn ends since 0.62+0.44 > 1 and x = 0.44.
If the second players draws 0.1, 0.27 and 0.91, the second player turn ends since 0.62+0.44+0.1+0.27+0.91 > 2 and y = 0.91. Since y > x, the second player wins.

Louise thinks about it for a second, and objects: "That's not fair".
What is the probability that the second player wins?
Give your answer rounded to 10 places behind the decimal point in the form 0.abcdefghij

# Not solved yet

My simple Monte-Carlo simulation isn't accurate enough. I need to come up with a smarter algorithm.

# What I've Done So Far

I wrote a simple Monte-Carlo simulation which finds the first 4 digits in a few seconds: 0.5276

My current thinking is that I could implement a better numerical approximation of the probability of drawing x as the last number (that is, some p(x)).
Julie's number would be p(0) and Louise's p(p(0)). The result is the probability that p(0) < p(p(0)).

However, I'm not sure whether that would be enough to compute the first 10 digits reliably.

# Interactive test

This feature is not available for the current problem.

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       #include <iostream>

// a simple pseudo-random number generator, returning a value between 0 and 1 (exclusive)
// (produces the same result no matter what compiler you have - unlike rand() from math.h)
double myrand()
{
// based on code from problem 227, modified to return a value between 0 and 1 (exclusive)
static unsigned long long seed = 0;
seed = 6364136223846793005ULL * seed + 1;

const unsigned int ValidBits = 31; // maybe not all bits are random, but it's not a high-quality algorithm anyway
const unsigned int Scaling   = 1 << ValidBits;
auto result = ((seed >> 30) % Scaling) / double(Scaling);
return result;
}

double monteCarlo(unsigned int iterations)
{
// count how often player 2 wins
unsigned int twoWins = 0;

for (unsigned int i = 0; i < iterations; i++)
{
// sum of all random numbers
double total = 0;
// last random number of player 1
double lastOne = 0;
while (total < 1)
{
lastOne = myrand();
total += lastOne;
}

// last random number of player 1
double lastTwo = 0;
while (total < 2)
{
lastTwo = myrand();
total += lastTwo;
}

if (lastTwo > lastOne)
twoWins++;
}

return twoWins / double(iterations);
}

int main()
{
// the first 4 digits should be correct - but I have to find the first 10 digits, which is impossible with Monte-Carlo
while (true)
std::cout << monteCarlo(100000000) << std::endl;
return 0;
}


This solution contains 8 empty lines, 8 comments and 1 preprocessor command.

# Changelog

November 2, 2017 wrote Monte-Carlo simulation

# Difficulty

Project Euler ranks this problem at 55% (out of 100%).

# Heatmap

Please click on a problem's number to open my solution to that problem:

 green solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too yellow solutions score less than 100% at Hackerrank (but still solve the original problem easily) gray problems are already solved but I haven't published my solution yet blue solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much orange problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte red problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too black problems are solved but access to the solution is blocked for a few days until the next problem is published [new] the flashing problem is the one I solved most recently

I stopped working on Project Euler problems around the time they released 617.
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The 310 solved problems (that's level 12) had an average difficulty of 32.6% at Project Euler and
I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.

My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.

Look at my progress and performance pages to get more details.

 << problem 429 - Sum of squares of unitary divisors Powers With Trailing Digits - problem 455 >>
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