**Toolbox:**

- In a toss of 3 coin .X denotes the number of tails
- X can take value X=0(notail);(X=1)(1 tail)
- X=2(2tails);X=3(3tails)
- \(x_i\)=i-1,2,3,
- \(P_i\)=are there correrpending proabilities
- =\(\sigma\;\Large\;\sqrt{\sum_iP_i\;x_i^2\;-\;\sum_i\;P_i\;x_i)^2}\)
- Probability of getting 1 head in a single throw=\(\large\frac{1}{2}\);tail=\(\large\frac{1}{2}\)

P(x=0)=P(no tail)=\(P(H\;H\;H)\)

=\(\Large\frac{1}{2}\;\times\;\frac{1}{2}\;\times\;\frac{1}{2}\;=\;\frac{1}{8}\)

P(X=1)=P(1tail)

\(P(T\;T\:H\:,T\;H\;H\;,H\;T\;T)\)

=\(\Large\frac{1}{8}\;+\;\frac{1}{8}\;+\;\frac{1}{8}\;=\;\frac{3}{8}\)

P(X=2)=P(2tail)

\(P(H\;T\;T\;,T\;H\;T\;,T\;T\;H)\)

\(\Large\frac{1}{8}\;+\;\frac{1}{8}\;+\;\frac{1}{8}\;=\;\frac{3}{8}\)

P(X=3)=P(3tail)

\(P(T\;T\;T)\)

=\(\Large\;\frac{1}{8}\)

\(\Large\;\sum\;P_iX-i\;=\;\frac{1}{8}\;\times0+\;\frac{3}{8}\;\times1+\;\frac{3}{8}\;\times2+\;\frac{1}{8}\;\times3\)

\(\Large\;\frac{12}{8}\)

\(\Large\;\sum\;P_i\;X_i^2\;=\;\frac{1}{8}\;\times0+\;\frac{3}{8}\;\times1+\;\frac{3}{8}\;\times4+\;\frac{1}{8}\;\times9\)

\(\Large\frac{42}{8}\)

\(\Large\;\sigma=\sqrt{\frac{42}{8}-(\frac{12}{8})^2}\)=\(\Large\sqrt\frac{192}{64}\)

\(\Large\frac{\sqrt3}{2}\)