probability of winning if you switch is 1 - (1 / n) aka (n-1)/n. probability of winning if you don't is 1 / n. By switching, you are simply betting that your

probability of winning if you switch is 1 - (1 / n) aka (n-1)/n. probability of winning if you don't is 1 / n. By switching, you are simply betting that your original guess of 1/n was wrong.