Exactly 1 nonconforming unit = 0.369458128
1 or more nonconforming units = 0.433497537
Due to formatting issues, I'll use the notion C(n,x) for N choose X. Which would be n!/(x!(n-x)!).
Now for the case of exactly 1, the probability will be:
P = C(3,1)*C(30-3,4)/C(30,5)
To explain it, you choose exactly 1 nonconforming item out of the 3 possible, then fill the sample with 4 more conforming items to complete the sample size of 5. Finally, you divide by the number of different ways you can select 5 items out of the entire group of 30. So doing the math, you get P = C(3,1)*C(30-3,4)/C(30,5) = 3*17550/142506 = 0.369458128 = 36.9458128%
For the case of 1 or more nonconforming units you do the sum of x ranging from 1 to 3 with the formula
C(3,x)*C(30-3,5-x)/C(30,5)
or you can set x to 0 and evaluate
1 - C(3,0)*C(30-3,5-0)/C(30,5)
Let's do it both ways.
C(3,1)*C(30-3,5-1)/C(30,5) +
C(3,2)*C(30-3,5-2)/C(30,5) +
C(3,3)*C(30-3,5-3)/C(30,5)
= 3*17550/142506 + 3*2925/142506 + 1*351/142506
= 0.369458128 + 0.061576355 + 0.002463054
= 0.433497537
And doing it the "simple" way
1 - C(3,0)*C(30-3,5-0)/C(30,5)
= 1 - 1*80730/142506 = 1 - 0.566502463 = 0.433497537
