examday

/TMUA

Each use of either of two machines produces one of the numbers $0$, $1$ and $2$, each with positive probability. The fixed probabilities may be different for the two machines. Let $X$ and $Y$ be the results of independent uses of the first and second machines.

The two probability-weighted means are equal:

It is also given that

Consider the following statements.

1. If $\Pr(X=0)\ne\Pr(Y=0)$, then $m=1$.
2. $\Pr(X=Y)\geq\dfrac13$.
3. If $\Pr(X=0)=\Pr(X=2)$, then $\Pr(Y=0)=\Pr(Y=2)$.

Which of these statements must be true?