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एक समुच्चय $S$ में 7 अवयव हैं। $S$ का एक अरिक्त उपसमुच्चय $A$ तथा $S$ का एक अवयव $x$, यादृच्छया चुने गए, तो $x \in A$ की प्रायिकता है
$\frac{1}{2}$
$\frac{64}{127}$
$\frac{63}{128}$
$\frac{31}{128}$
Solution
Let $\mathrm{S}=\left\{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}\right\}$
Let the chosen element be $x_{i}$,
Total number of subsets of $S=2^{7}=128$
No. of non-empty subsets of $S=128-1$ $=127$
We need to find number of those subsets that contains $x_{i}$,
$\boxed2\boxed2\boxed2\boxed2\boxed1\boxed2\boxed2$
${x_1}\,{x_2} – – {x_i} – – {x_7}$
For those subsets containing $x_{i}$ each element has 2 choices.
i.e., (included or not included) in subset,
However as the subset must contain $x_{i}$ $x_{f}$ has only one choice. (included one)
So, total no. of subsets containing $x_{i}=2 \times 2 \times 2 \times 2 \times 1 \times 2 \times 2=64$
Required prob
$ = \frac{{No.\,\,of\,\,subsets\,\,containing\,\,{x_i}}}{{Total\,\,no.\,\,of\,non – empty\,\,subsets}}$
$=\frac{64}{127}$
