Project Euler problem 323

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At first glance this question is bit computing, in fact, a probability distribution and related topics.

There y0 y1 y2 series 32bits unsigned number
x is initially 0, then the series and the number of exclusive-or operation, the operation N times, until x is 2 ^ 32-1, in fact, all bits are 1 x.
N demand expectations.

How to calculate expectations?
For each n is calculated corresponding to the probability p (n), then the weighted sum.
We now look one by one analysis
N is 1, then that y0 every bits are 1, the probability is 1 / (2 ^ 32)
N 2, were calculated for each bit, the first bit, the result is the probability that 1 minus the probability is not 1, the probability is not 1 is the first bit is not 0, the first bit y0 y1 is not 0, the probability is 1 / (2 ^ 2), then p (2) = (1-1 / (2 ^ 2)) ^ 32 - P (. 1)
N is 3, p (3) = ( 1-1 / (2 ^ 3)) ^ 32 - p (1) - p (2)
so
topic after ten decimal required, if the p (n) is small, the results are not affected by multiplying n, that is, p (n) * 10 ^ n is less than ten.

When I do a calculation of the probability of a data structure stored in the numerator and denominator. After the answer I came out, looked at the analysis of others, feel the direct use double to save the probability can give the correct answer, I did not try, uncertain.
Here is my code:

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public class { public BigInteger Numerator { get; set; } public BigInteger Denominator { get; set; } public (BigInteger numerator, BigInteger denominator) { Numerator = numerator; Denominator = denominator; } Big Box   Euler Project 323 title = "line">} public static string GetAnswer() { List<Probability> probabilities = new List<Probability>(); probabilities.Add( new Probability( 0, 1)); BigInteger twoPow32 = BigInteger.Pow( 2, 32); for ( int i = 1; ; i++) { BigInteger baseDenominator = BigInteger.Pow( 2, i); BigInteger baseNumerator = baseDenominator - 1; BigInteger lastBaseDenominator = BigInteger.Pow( 2, i - 1); BigInteger lastBaseNumerator = lastBaseDenominator - 1; BigInteger denominator = BigInteger.Pow(baseDenominator, 32); BigInteger numerator = BigInteger.Pow(baseNumerator, 32); BigInteger lastNumerator = BigInteger.Pow(lastBaseNumerator, 32); probabilities.Add( new Probability( numerator - lastNumerator * twoPow32, denominator)); if (probabilities[i].Denominator / probabilities[i].Numerator / i > BigInteger.Pow( 10, 12)) { break; } } BigInteger ansDenominator = probabilities.Last().Denominator; BigInteger ansNumerator = 0; for ( int i = 1; i < probabilities.Count; i++) { Probability current = probabilities[i]; BigInteger times = ansDenominator / current.Denominator; ansNumerator += times * current.Numerator * i; } BigInteger times11 = ansNumerator * 100000000000 / ansDenominator; double answer = ( double)times11 / 100000000000; return answer.ToString( "N10"); }