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// Multinomial formula | |
def multi(ks: Vector[Int]): Double = { | |
(1 to ks.sum) | |
.map(_ / ks.size.toDouble) | |
.zip(ks.flatMap(1 to _)) | |
.map({ case (n, d) => n / d }).product | |
} | |
// Expected number of cereal boxes you have to buy if you want to collect at least `m` of each of `n` surprise mystery toys | |
def e(n: Int, m: Int, M: Int = 20, v: Vector[Int] = Vector.empty): Double = { | |
if (n == 1) multi(v :+ (m-1)) * (v.sum + m) | |
else (m to M).map(i => e(n-1, m, N, v :+ i)).sum | |
} | |
// Same thing but done by random sampling | |
def g(n: Int, m: Int): Distribution[Double] = { | |
discreteUniform(1 to n) | |
.until(xs => xs.groupBy(x => x).values.map(_.size).count(_ >= m) == n) | |
.map(_.size.toDouble) | |
} | |
// Special case of m=1 | |
def f(n: Int) = (1 to n).map(i => 1.0 * n / i).sum | |
// e reproduces f when m=1 | |
scala> f(3) | |
res0: Double = 5.5 | |
scala> e(3, 1, 20) | |
res1: Double = 5.499903100004326 | |
// e matches g | |
scala> g(5, 4).ev | |
res2: Double = 32.897099999999995 | |
scala> e(5, 4, 20) | |
res2: Double = 32.44101314690209 | |
scala> g(2, 10).ev | |
res3: Double = 23.536799999999964 | |
scala> e(2, 10, 50) | |
res4: Double = 23.523939115154043 |
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