13 % of people took a math course. What is the probability that in a 350 randomly selected sample, less than 40 people take the course?

So I have $X\sim Bin(350,\frac{13}{100})$

Then let $Y\sim N(\frac{13}{100},{\sqrt{0.00032314}}^{2})$

I want $P(X<40)=P(\hat{p}<\frac{4}{10})$

Then $P(z<\frac{\frac{4}{10}-\frac{13}{100}}{{\sqrt{0.00032314}}^{2}})$ which is obviously very wrong.

Where did it go wrong, and why did it go wrong?

So I have $X\sim Bin(350,\frac{13}{100})$

Then let $Y\sim N(\frac{13}{100},{\sqrt{0.00032314}}^{2})$

I want $P(X<40)=P(\hat{p}<\frac{4}{10})$

Then $P(z<\frac{\frac{4}{10}-\frac{13}{100}}{{\sqrt{0.00032314}}^{2}})$ which is obviously very wrong.

Where did it go wrong, and why did it go wrong?