$P(A') = \frac{{999}}{{1000}} \times \frac{{998}}{{999}} \times ..... \times \frac{{1000 - k + 1}}{{1000 - k + 2}} \times \frac{{1000 - k}}{{1000 - k + 1}}$
$ \Rightarrow $$P(A')$$ = \frac{{1000 - k}}{{1000}}$
$⇒$ $P(A) = 1 - \frac{{1000 - k}}{{1000}} = \frac{k}{{1000}}.$
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| Column $I$ | Column $II$ |
| $(A)$ Circle | $(p)$ The locus of the point $(h, k)$ for which the line $h x+k y=1$ touches the circle $x^2+y^2=4$ |
| $(B)$ Parabola | $(q)$ Points $z$ in the complex plane satisfying $|z+2|-|z-2|= \pm 3$ |
| $(C)$ Ellipse | $(r)$ Points of the conic have parametric representation $x=\sqrt{3}\left(\frac{1-t^2}{1+t^2}\right), y=\frac{2 t}{1+t^2}$ |
| $(D)$ Hyperbola | $(s)$ The eccentricity of the conic lies in the interval $1 \leq x<\infty$ |
| $(t)$ Points $z$ in the complex plane satisfying $\operatorname{Re}(z+1)^2=|z|^2+1$ |