Mathematics Asked by Buyanov Igor on December 9, 2020

Suppose, I have two random variables $X$ and $Y$. Both of them can produce output 0 and 1. Variable $X$ has the probability of emitting 1 is $0.7$ and $Y$ has the probability $0.2$ (so the probability of emitting 0 is $0.8$). The variables are independent.

I try to figure out the way, how I can combine these variables such that the output of a combination converges to the $0.5$ of probability of emitting 0 (and 1, respectively).

Please, help.

EDIT.

Under the "combining" I mean like how I should sequentially "call" these variables that in this sequence an quantities of 0 and 1 roughly be equal.

We want the probability to be around $0.5$. If there are $x$ instances of $X$ (for example $x = 2$ when $X$ occurs two times) and $y$ instances of $Y$, then the probability is $binom{x+y}{x} cdot 0.7^x cdot 0.2^y$.

(We have a $0.7^x$ and $0.2^y$ chance for each of $X$ and $Y$, but we also multiply by $x+y$ choose $x$ because we can order the probabilities in any way. For example, there are two ways to choose X and Y 1 time: XY or YX.)

$binom{x+y}{x} cdot 0.7^x cdot 0.2^yapprox 0.5$, so we need to find some values of $x$ and $y$ that roughly work out. Unfortunately, no integer solution comes close to $0.5$ (according to Wolfram and Desmos); the closest possible is when $x = 2$ and $y = 0$.

Therefore you can combine $2$ of $X$ and $0$ of $Y$ to get around $0.5$.

-FruDe

Answered by FruDe on December 9, 2020

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