Let $ n$ be a positive integer congruent to $1$ modulo $4$. Xantippa has a bag of $n + 1$ balls numbered from $ 0$ to $n$. She draws a ball (randomly, equally distributed) from the bag and reads its number: $k$, say. She keeps the ball and then picks up another $k$ balls from the bag (randomly, equally distributed, without repossession). Finally, she adds up the numbers of all the $k + 1$ balls she picked up. What is the probability that the sum will be odd?