<< problem 284 - Steady Squares Quadtree encoding (a simple compression algorithm) - problem 287 >>

# Problem 286: Scoring probabilities

Barbara is a mathematician and a basketball player.
She has found that the probability of scoring a point when shooting from a distance x is exactly (1 - x/q),
where q is a real constant greater than 50.

During each practice run, she takes shots from distances x = 1, x = 2, ..., x = 50 and, according to her records,
she has precisely a 2 % chance to score a total of exactly 20 points.

# My Algorithm

My solution is based on Dynamic Programming and bisection search.

The function probability returns the chance that exactly 20 are made from distances 1,2,...,50 given a certain q.
And the main function iteratively narrows a range of potential values for q until the range is so small that I can be sure
to be extremely close to the true value of q (error is less than 10^-10).

The bisection part was obvious to me right from the start.
However, I needed a few minutes to figure out that I can use a Dynamic Programming approach (again :-) ) for probability.
I was worried that the cache size might grow too fast and tried to solve the problem analytically.
But as it turns out the cache contains only about 33000 values at the end (even without clearing it when a new q is processed).

## Alternative Approaches

My simple recursion could be probably replaced by an iterative algorithm because of the low number of different states (made, distance).

## Note

I play basketball and in my opinion the problem is phrased incorrectly:
according to the "Project Euler Basketball rules" Barbara gets only 1 point per shot she made.
In real life, a shot is usually worth 2 points. You get 1 point for a free throw and 3 points for a "long-range" shot.

The correct result is found after 39 iterations.
Due to the nature of bisection, narrowing the initial interval doesn't really speed up the process.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

Input data (separated by spaces or newlines):
Note: Enter the number of shots made.

This is equivalent to
echo 25 | ./286

Output:

Note: the original problem's input 20 cannot be entered
because just copying results is a soft skill reserved for idiots.

(this interactive test is still under development, computations will be aborted after one second)

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

       #include <iostream>
#include <iomanip>
#include <map>

// Barbara hits exactly 20 times out of 50 attempted shots
unsigned int threshold   = 20;
const unsigned int maxDistance = 50;
// Barbara has a 2% chance of making exactly 20 points (50 attempts)
const double chanceHitExactly  =  0.02;

// find probability of making 20 shots given a certain q
// note: second and third parameter are needed for internal recursion only
double probability(double q, unsigned int made = 0, unsigned int distance = 1)
{
return 0;

// finished ? => made exactly 20 shots ?
if (distance > maxDistance)
return made == threshold ? 1 : 0;

// memoize
static std::map<std::pair<unsigned int, double>, double> cache;
auto id = std::make_pair(made * (maxDistance + 1) + distance, q);
auto lookup = cache.find(id);
if (lookup != cache.end())
return lookup->second;

// compute probabilities
double chanceHit  = 1 - distance / q;
double chanceMiss = 1 - chanceHit; // can be simplified to distance/q

// go back one step
distance++;
// and attempt one more shot: could be a hit, could be a miss ...
double result = chanceHit  * probability(q, made + 1, distance) +

cache[id] = result;
return result;
}

int main()
{
// user-defined number of made shots from distances 1..50
std::cin >> threshold;

// q >   50 because otherwise there would be a negative probability for the shot from distance 50
double low  = maxDistance;
// q <= 100 is a pretty random choice ... and it's much larger than the true value q
double high = 100;

// ten decimal places
double accuracy = 0.0000000001;
// while interval is still too wide ...
while (high - low > accuracy)
{
// analyze its midpoint
auto mid = (high + low) / 2;

// below 2% ?
if (probability(mid) < chanceHitExactly)
high = mid; // mid is larger  than the true q
else
low  = mid; // mid is smaller than the true q
}

if (low > 50)
// low == high (at least with respect to their first 10 decimal places)
std::cout << std::fixed << std::setprecision(10) << low << std::endl;
else
std::cout << "impossible" << std::endl;

return 0;
}


This solution contains 13 empty lines, 19 comments and 3 preprocessor commands.

# Benchmark

The correct solution to the original Project Euler problem was found in 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
Peak memory usage was about 4 MByte.

(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL)

See here for a comparison of all solutions.

Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL.

# Changelog

July 6, 2017 submitted solution

# Difficulty

Project Euler ranks this problem at 50% (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 the flashing problem is the one I solved most recently

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The 289 solved problems (level 11) had an average difficulty of 32.1% at Project Euler and
I scored 13,486 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.

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