Everyone says $N(d_2)$ is the probability of the option being exercised but stocks that have really high volatility have really expensive options indicating a high likelihood of expiring in the money. If volatility is very high we see that s $N(d_2)$ goes to 0, so doesn't that contradict our first statement? Isn't $N(d_1)$ the true probability?
This is indeed a very basic question. Please have a look at the following paper which will answer all your questions in a very intuitive and step-by-step fashion: