How to construct an event with a given probability, based on another event Assume that an event $A$ takes places with probability $p$ in some random experiment.
Is it possible to construct from here an event with probability $(1-2p)^2$? To be more clear, I would be interested in a construction that is similar to one in the examples below.
Example 1:
Assume that we repeat the same experiment twice, in the same conditions. Then the probability of obtaining $A$ exactly one time is $2p(1-p)$.
Example 2 (with a concrete experiment):
Consider an urn with red and blue balls, where the probability of extracting a red ball at random is $p$. We extract two balls at random, one at a time, each time putting the ball back. If the first ball is red we repaint the balls by switching their color. Then the event of obtaining at least one red ball is $p(2-p)$ (if I made the computation correctly).
Edit: If constructing such an event is not possible, then it is also acceptable that the experiment be changed somehow (see Example 2). It is also acceptable if the probability $(1-2p)^2$ is a conditional probability.
 A: The famous paper A Bernoulli factory by M. S. Keane and George L. O'Brien (Journal ACM Transactions on Modeling and Computer Simulation,
Volume 4 Issue 2, 1994, pages 213-219) study, given a (known) function $f:[0,1]→(0,1)$, the problem of simulating a coin with probability of heads $f(p)$ by tossing $N$ times a coin with unknown heads probability $p$, where $N$ may be random. They show that such a simulation scheme with $P_p(N<\infty)=1$ exists if and only if $f$ is continuous. 
The problem with your function, namely, $f(p)=|1-2p|$, is its zero at $p=\frac12$. Thus, every admissible simulation scheme must produce tails almost surely, when $p=\frac12$. In particular, on each $[N\leqslant n]$, one must get tails almost surely. Since $P_{1/2}$ and $P_p$ are equivalent on each cylinder $\{0,1\}^n$, one must get tails almost surely when the bias is $p$ as well. 
To sum up, it is impossible to simulate $f(p)=|1-2p|$ on $p=\frac12$ and on another value $p=p^*$ with $p^*\ne\frac12$, and a fortiori on every larger set of values of $p$, because if the simulation scheme works when $p=\frac12$, then under $P_{p^*}$ one simulates $\tilde f(p^*)=0$ instead of $f(p^*)=|1-2p^*|$.
A: Let me ask a (maybe?) simpler question: Given a coin, which gets heads with probability $p$, what kind of probabilities are 'constructible'? To be more concrete: Define a function $f(p)$ to be a constructible probability if there exists an experiment where I flip the coin $n$ times, take some event $E$ which is a subset of all the possible outcomes, and the probability of $E$ is equal to $f(p)$. 
First, the probability of any specific sequence with $k$ heads and $n - k$ tails is going to have probability $p^k(1 - p)^{n - k}$. Since any event is a union of these 'simple' events, it follows that any constructible probability can be written as a polynomial in $p$ with integer coefficients. More explicitly, the constructible probabilities are exactly sums of positive integer multiples of expressions like $p^a(1 - p)^b$
However, $p^a(1 - p)^b$ is positive whenever $0 < p < 1$, and it follows that sums of positive integer multiples of such expressions are also positive when $0 < p < 1$. Hence, all constructible probabilities satisfy this. But then $(1 - 2p)^2$ is not constructible, since it equals zero when $p = 1/2$.
