{"conversations": [{"from": "user", "value": "Escriviu una prova, usant LaTeX, mostrant que escolliu dos enters aleatoris i multipliquen els resultats junts en un nombre incl\u00fas amb probabilitat de $\\ frac{ 3} {4}."}, {"from": "assistant", "value": "Permet que $A$ sigui l' esdeveniment que dos enters seleccionats a l' atzar siguin igualment, i deixar que $B$ sigui l' esdeveniment que dos enters seleccionats a l' atzar s\u00f3n rars. Volem trobar la probabilitat de l' esdeveniment $C = A\\\\ cup B$, on $C$ \u00e9s l' esdeveniment de dos enters seleccionats aleat\u00f2riament \u00e9s fins i tot. Tenim:\\\\ begin{ align * {g * {\\\\ 1} &\\\\ cg{ 4}\\\\ t\\\\ ct\\\\ {c\\\\ 2} i\\\\ {c\\\\ {c\\\\ {c\\\\ {c\\\\ 2} {c\\\\ {c\\\\ 2} i\\\\ {c\\\\ {c\\\\ {c\\\\ 4} {c\\\\ {c\\\\ 4} & {c\\\\ {c\\\\ {c\\\\ {c\\\\ {c\\\\ {cg\\\\ {} i\\\\ {c\\\\ {c\\\\ {c\\\\ {c\\\\ {c\\\\ {\\\\ {cg\\\\ {\\\\ {} i\\\\ {cg\\\\ {}.\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ {\\\\ {\\\\ {\\\\ {c\\\\ {cg\\\\ {\\\\ {cggggggg\\\\ {} i\\\\ {\\\\ {\\\\ {}.\\\\ {\\\\ 2} i\\\\ 2} i\\\\ 2} i\\\\ {} i\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {c\\\\ {\\\\ {\\\\ {\\\\ {c\\\\ {\\\\ {c\\\\ {c\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ {\\\\ {c\\\\ {\\\\ {\\\\ {cggggg\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ 2} i\\\\ 2}} i\\\\ 2} i\\\\ {\\\\ 2\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ {\\\\ 2} i\\\\ 2} i\\\\ {c\\\\ {\\\\ {cgggg\\\\ {\\\\ {cgggggggg\\\\ {\\\\ {\\\\ 2} i\\\\ 2} i\\\\ 2} i\\\\ {\\\\ 2} i\\\\ {"}]}