Question

Se considera multimea A={1,2,3,4,5,6}. Determinati probabilitatea ca, alegand la intamplare una dintre submultimile nevindecabile ale lui A, aceasta sa contina cel puțin trei elemente.

Answer 1

   
[tex]\displaystyle\\ A=\{1,2,3,4,5,6\}\\ \text{Card } A = 6\\ \texttt{Calculam numarul de multimi.}\\\\ C_6^1 = 6~\text{ submultimi cu 1 element}\\\\ C_6^2 = \frac{6\times 5}{1\times 2} = 3 \times 5 = 15~\text{ submultimi cu 2 elemente}\\\\ C_6^3 = \frac{6\times 5\times 4}{1\times 2 \times 3} = 5 \times 4 = 20~\text{ submultimi cu 3 elemente}\\\\ C_6^4 = C_6^2 = 15~\text{ submultimi cu 4 elemente}\\\\ C_6^5=C_6^1 = 6~\text{ submultimi cu 5 elemente}\\\\ C_6^6 = 1~\text{ submultime cu 6 elemente}[/tex]

[tex]\displaystyle\\ \text{Total } ~~6 + 15 + 20 + 15 + 6 + 1 = 63 ~\text{submultimi nevide}\\\\ \text{Din care } ~20 + 15 + 6 + 1 = 42 ~\text{ de multimi c cel putin 3 elemente}\\\\ p = \frac{\text{Cazuri favorabile}}{\text{Cazuri posibile}}= \frac{42}{63} = \boxed{\bf \frac{2}{3}} = \boxed{\bf 0,(6)} = \boxed{\bf 66,(6)\% } [/tex]