Accurate definition of BPP
Repeating what MCH said, every time you run the algorithm, you use fresh random bits.

As an example, here is a BPP algorithm that computes the number $1$:

Toss a biased coin (heads w.p. 2/3, tails w.p. 1/3). If heads, output 1, else output 0.


The algorithm "succeeds" with probability $2/3$. The amplification procedure amounts to tossing $n$ biased coins, outputting $1$ if the majority of them were heads, and $0$ otherwise. Since we expect roughly $(2/3)n &gt; n/2$ of the coins to be heads, it is very likely that the amplified algorithm is successful (outputs $1$).

In other words, when you run your algorithm $n$ times, each time you run it, you re-toss all the coins. So the amplified algorithm tosses $n$ times as many coins, and the probability is taken over all these coin tosses.