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> $3$ of $10$ products are defective. If $5$ products are selected at random, what is the probability that all sample contains all three defective products
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I have already calculated $P(X=3)= \dfrac1{12}$ and $P(X\geq1)=\dfrac1{12}$, where $X=$"Product is defective". After that, I have formed expression of conditional probability

$$P(P=3｜P\geq1)= \dfrac{P(P(X=3)\,\mathrm{and}\,P(X\geq1))}{P(X>=1)}$$

This is where I am stuck. The solution is $$\dfrac{P(X=3)}{1-P(X=0)}$$ but I have no idea where that $P(X\geq1)$ disappears from numerator $P(P(X=3) and P(X\geq1))$.

Thanks for your help!