• From all families with two children, at least one of whom is a boy, a family is chosen at random. This would yield the answer of 1/3.
  • From all families with two children, one child is selected at random, and the sex of that child is specified to be a boy. This would yield an answer of 1/2.

The subtle difference between those two interpretations from https://en.wikipedia.org/wiki/Boy_or_Girl_paradox was hard for me to grasp, so I wrote some code to see it myself.

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#include <random>
#include <iostream>

using namespace std;

enum gender {
  B,
  G,
};

gender db[4][2] = {
  {B, B},
  {B, G},
  {G, B},
  {G, G},
};

constexpr int count = 100*1000;

int total;
int found;

void f() {
  random_device rd;
  mt19937 gen(rd());
  uniform_int_distribution<> array_index_gen(0, 3);
  uniform_int_distribution<> sub_index_gen(0, 1);

  total = found = 0;
  for (auto n = 0; n < count; ++n) {
    auto i = array_index_gen(gen);
    if (db[i][0] == B || db[i][1] == B) {
      total++;
      if (db[i][0] == db[i][1]) {
        found++;
      }
    }
  }

  cout << "found/total: " << found << "/" << total
    << " = " << found * 100.0/total
    << endl;
}

void g() {
  random_device rd;
  mt19937 gen(rd());
  uniform_int_distribution<> array_index_gen(0, 3);
  uniform_int_distribution<> sub_index_gen(0, 1);

  total = found = 0;
  for (auto n = 0; n < count; ++n) {
    auto i = array_index_gen(gen);
    auto j = sub_index_gen(gen);
    if (db[i][j] == B) {
      total++;
      if (db[i][1-j] == B) {
        found++;
      }
    }
  }

  cout << "found/total: " << found << "/" << total
    << " = " << found * 100.0/total
    << endl;
}

int main()
{
  f();
  g();

  return 0;
}

The output is

1
2
found/total: 25184/74954 = 33.5993
found/total: 25076/50259 = 49.8936

In summary, how we came by this piece of info, “at least one is a boy”, plays a vital role in the final result.

§references