**Toolbox:**

- \(A\;pack\;of\;52\;cards\;contain\;4\;aces\)
- \(In\;a;toss\;of\;dies\;twice\)
- \(X=\;no\;of\;aces\)
- \(X=(0\;1\;2\;)\)
- \(Mean\;=\sum\;P_i\;X_i\)
- \(\sigma\;=\sqrt{variance}=\sqrt{\sum\;P_iX^2_i\;-\;(\sum\;P_iX_i)^2}\)

\(P(X=0)=p(no\;aces)\)

\(\Large\frac{48}{52}\times\frac{47}{51}\)

\(P(X=1)=p(1\;aces/no\;aces)\)

\(\Large\;2(\frac{48}{52}\times\frac{47}{51})\)

\(P(X=2)=p(both\;aces)\)

\(\Large\frac{4}{52}\times\frac{3}{51}\)

\(E(X)=\sum\;P_iX_i\)

=\(\Large\;0\times\frac{48\times47}{52\times51}\;+\;1\times2\times\frac{4}{52}\times\frac{48}{51}+2\times\frac{4\times3}{52\times51}\)

=\(\Large\;\frac{204}{26\times51}\;=\frac{2}{13}\)

\(\sum\;P_iX_i^2=\Large\;1^2\;\times\;2\;\times\;\frac{4}{52}\times\frac{48}{51}+2^2\times\frac{4\times3}{52\times51}\)

\(var\;=\;0.2952-(.153)^2\)

=\(0.1421\)

\(\sigma\;=\sqrt{0.1421}=0.377\)