Sum of probabilities of events vs. probability of at least one event $P_i$ is the probability of one event. As defined below, $a$ is the sum of all probabilities of events (that may or may not be independent), and $c$ is the overall probability that at least one event will happen. 
$a=\sum_{i=0}^n P_i $
$c=(1-\prod_{i=0}^n (1-P_i))$
$0\leq P_i \leq 1$
Take two events with probability of $0.5$ and $0.2$ for example: 
$$ a= 0.5 +0.2=0.7~, \qquad c = 1 - 0.5 \cdot 0.8=0.6$$

Is it true $\dfrac{a}{c}$ will increase as $n$, the number of $P_i$, increases? 

It seems true to me based on sample results, but I can't think of a formal proof.
 A: Yes, the question you meant to ask is true, although (as pointed out already) it is not quite correct as stated. To keep things clear, I will start by restating your question in a more precise form.
Let $n> 1$ be an integer and let $a_1,\ldots,a_n$ be positive real numbers, all between $0$ and $1$. Prove that
$$
\frac{\sum_{i=1}^{n-1}a_i}{1-\prod_{i=1}^{n-1}(1-a_i)}\leq \frac{\sum_{i=1}^{n}a_i}{1-\prod_{i=1}^{n}(1-a_i)}.
$$
Proof. Let $S=\sum_{i=1}^{n-1}a_i$ and let $P=\prod_{i=1}^{n-1}(1-a_i)$. Since the $a_i$ are positive, both denominators are also positive so we can multiply through and substitute $S$ and $P$ to obtain the equivalent inequality
$$
\frac{S}{1-P}\leq \frac{S+a_n}{1-(1-a_n)P}\iff S\bigl(1-(1-a_n)P\bigr)\leq (1-P)(S+a_n).
$$
Expanding this out and collecting like terms leads to the equivalent inequality
$$
a_nSP \leq a_n(1-P),
$$
which (since $a_n$ is positive) is equivalent to $SP\leq 1-P$, which is the same as $P(1+S)\leq 1$.
But this is true, since
$$
P(1+S)=\prod_{i=1}^{n-1}(1-a_i) \ \cdot \ \Bigl(1+\sum_{i=1}^{n-1}a_i\Bigr)\leq \prod_{i=1}^{n-1}(1-a_i)\ \cdot \ \prod_{i=1}^{n-1}(1+a_i)=\prod_{i=1}^{n-1}(1-a_i^2)\leq 1.
$$
This establishes the desired inequality. $\square$
