Solutions to Problems 1261-1270

Q1261 A hat contains $N = 2n$ tickets, $n = 2, 3, 4, \ldots,$ each marked with a number from $1$ to $N$. (Each ticket has a different number.) In a game, players are asked to draw from the hat two tickets, read them, and replace them. Prize winners are those who draw two numbers whose ratio is $2$. The money is refunded if the ratio is $2k$ for $k = 2, 3, \ldots$. Find the probability of

  1. a win;
  2. a refund.