Question

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

Answer 1

Number of subset of S $=2^{\left(|S|\right)}=2^7=128$

There are ${}^7{C}_n$ subsets of S having n elements

therefore,

$P(|A|=n)=\frac{{}^7{C}_n}{2^7}$

And, the combination are

${}^7{C}_0=1$

${}^7{C}_1=7$

${}^7{C}_2=21$

${}^7{C}_3=35$

${}^7{C}_4=35$

${}^7{C}_5=21$

${}^7{C}_6=7$

${}^7{C}_7=1$

$P(x\in A)=0+\left(\frac{1}{7}\right)\left(\frac{7}{128}\right)+\left(\frac{3}{7}\right)\left(\frac{35}{128}\right)+\left(\frac{4}{7}\right)\left(\frac{35}{128}\right)+\left(\frac{5}{7}\right)\left(\frac{21}{128}\right)+\left(\frac{7}{7}\right)\left(\frac{1}{128}\right)$

$P(x\in A)=\frac{1}{128}(0+11+6+15+20+15+6+1)$

$P(x\in A)=\frac{1}{128}\times 64$

$P(x\in A)=\frac{1}{2}$