Possibility for same Private Key Generation in identity-based encryption In private key generation of IBE scheme, private key is generated as follows: $SK_{ID}=s.H(ID)$ 
Suppose two servers use the same curve domain parameters.
If PKG on server A generates private key $SK_{ID}=s.H(ID)$
PKG on server B generates private key $SK_{ID}=b.H(ID)$
for the same ID.
If there are more than 100000 servers, will there be possibility for generating same private key?
I assume $s$ and $b$ are the main master secret key components of two IBE servers.

If $s=b$, then $SK_{ID,s}=SK_{ID,b}$ provided that the curve parameters and the hash function $H$ are the same. Now, the question is what's the probability of two IBE servers generating the same master secret key components. This depends on the actual scheme and the field it is defined over. 

In IBE, mostly pairing-based groups are used. We usually have $s,b\in \mathbb{Z}_{r}$ where the curve is defined over $\mathbb{F}_q$. For example, $r$ is set to be a solinas prime for Type A pairing in PBC. The default values for Type A pairing are $|q|=512$ and $|r|=160$. So, the probability of at least two of $n$ servers generating the same master secret key is essentially the Birthday Problem and can be computed as such: 

$$\begin{eqnarray}
p(n) &amp; = &amp; 1 - \frac{r!}{r^n\left(r-n\right)!}\\
&amp; = &amp; 1 - \frac{\prod^r_{i=r-n+1}i}{r^n}\\
&amp; = &amp; 1 - \prod^r_{i=r-n+1}\frac{i}{r}
\end{eqnarray}$$

The probability is roughly $3\cdot 10^{-39}$ in your case. Here's the groovy script I calculated it with. The result is $3\cdot 10^{-42}$ under the assumption that $|r|=170$.