Finding the PMF of conditional probability, poisson process. Don't understand where $10^6$ goes "Customers arrive at a bank according to a Poisson process with rate 6 per hour.
State (together with a proof) clearly the (conditional) probability mass function of the numbers of customers arrived during the first 20 minutes, given that 10 customers have arrived during the first hour."
I did this:
$$
P(X_{\frac{1}{3}} = x | X_1 = 10)
$$
From the definition of joint probability, we know
$$
P(X_{\frac{1}{3}} = x , X_1 = 10) = P(X_{\frac{1}{3}} = x | X_1 = 10) P(X_1 = 10)$$
Re-aranging gives:
$$
P(X_{\frac{1}{3}} = x | X_1 = 10) = \frac{ P(X_{\frac{1}{3}} = x , X_1 = 10)} {P(X_1 = 10)}$$
Let's first calculate $P(X_{\frac{1}{3}} = x , X_1 = 10)$:
This becomes:
$$
P(X_{\frac{1}{3}} = x , X_{\frac{2}{3}}= y) = P(X_{\frac{1}{3}} = x) P(X_{\frac{2}{3}} = y) \hspace{1cm} y = 10 - x $$
Using the Poisson formula, you get:
$$P(X_{\frac{1}{3}} = x) = \frac{e^{-2}2^x}{x!} \hspace 2cm P(X_{\frac{2}{3}}= y) = \frac{e^{-4}4^y}{y!}$$
And so 
$$
P(X_{\frac{1}{3}} = x , X_{\frac{2}{3}}= y) = e^{-6} \frac{2^x4^y}{x!y!}
$$
Also, using the Poisson formula, we get:
$$
P(X_1 = 10) = \frac{e^{-6}6^{10}}{10!}
$$
So 
$$\frac{ P(X_{\frac{1}{3}} = x , X_1 = 10)} {P(X_1 = 10)} = \frac{e^{-6} \frac{2^x4^y}{x!y!}}{\frac{e^{-6}6^{10}}{10!}}
$$
Which I get to be
$$
\frac{10! 2^x 4^y}{x!y! 6^{10}}$$
But in the answers, it says it should be
$$
\binom{10}{x}  \left( \frac{1}{3} \right)^x \left( \frac{2}{3} \right)^{10 - x}
$$
Why is this?

I assume in your question you mean $6^{10}$ as opposed to $10^6$. So, let's start with what you have:
$$\frac{10! 2^x 4^{10-x}}{x!(10-x)!6^{10}} = \frac{10!}{x!(10-x)!}\times\frac{2^x 4^{10-x}}{6^{10}}.$$
The first term is $\binom{10}{x}.$ Let's pull $2^{10}$ out of the top and bottom of the second term and cancel them, leaving us with
$$\binom{10}{x} \frac{1^x 2^{10-x}}{3^{10}} = \binom{10}{x} \frac{1^x}{3^x} \frac{2^{10-x}}{3^{10-x}} = \binom{10}{x}\left(\frac13\right)^x\left(\frac23\right)^{10-x}$$
which is the answer you want.
