$$ p(S,t;S',t') = \frac{1}{\sigma S'\sqrt{2\pi (t'-t)}} \exp\left(-\frac{(\log(S/S') + (\mu-1/2\sigma^2)(t'-t))^2}{2\sigma^2(t'-t)}\right) $$
I found this equation when I was reading "Paul Wilmots on Quantitative Finance" which calculates the probability that of a stock price ending/landing on a particular price (S'). So if the stock price is 100 USD and the volatility is 30%, the probability the stock closes at 105 USD, exactly 30 days from now, according to the formula, is 3.850%.
When I tried to use the formula for different strikes S' (95 96 97 98 99 100 101 102 103..), integer wide, and a fixed 30 days, the probabilities at those respective strikes summed to exactly 1, as it should. However when I used a fractional spacing/mesh (100 100.1 100.2 100.3 ...) I the probabilities summed more than 1. And that makes sense I guess? If the probability that a stock worth a 100 USD now will close at 105 USD, 30 days from now, is 3.85% than it shouldn't vary too far from the the probability of a 100 USD stock closing at 105.10 USD 30days, the two numbers are within the vicinity of one another by 10 cents.
But is there a way to normalize the formula such that if I wanted to know the probability that a stock will land within the interval [105.00, 106.00], I would not be getting probabilities that sum to more than 1?