Is $\prod_{i\in I}x_i$ equal to $1-\prod_{i\in I}(1-x_i)$?

Question

Are these 2 equations equivalent? If so how can I prove it, if not how can I disprove it?

$CI(t)=\prod_{t_i<t}(d_i/n_i)$

$CI(t)=1-\prod_{t_i<t}(1-(d_i/n_i))$

Answer 1

Just try it:

$$ab \stackrel{?}{=} 1 - (1-a) \cdot (1-b)$$

Well:
$1 - (1-a) \cdot (1-b) = 1 - (1 - a - b + ab) = a + b - ab \ne ab$

So, no. Those equations are not equal.