Question

A set $S$ contains $7$ elements. A non-empty subset $A$ of $S$ and an element $x$ of $S$ are chosen at random. Then the probability that $x\in A$ is:

• A. $\frac{1}{2}$
• B. $\frac{64}{127}$
• C. $\frac{63}{128}$
• D. $\frac{31}{128}$

Answer 1

Answer: (B)

Let $S=\left\{x_1,x_2,x_3,x_4,x_5,x_6,x_7\right\}$
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 $x_i$.

$x_1{x}_2-------x_i----x_7$
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 $x_i,x_i$ has only one choice. (included one)
So, total no. of subsets containing
$x_i=2\times 2\times 2\times 2\times 1\times 2\times 2=64$
Req. prob. $=\frac{\text{ No. of subsets containing }{x}_i}{\text{ Total no. of non-empty subsets }}=\frac{64}{127}$