**Toolbox:**

- let\((E_1)\)be the event 1st marble is red
- \((E_2)\)at least one is black
- First marble is red can occur in following ways
- \((E_1)\)={\(R\;B\;B)(R\;B\;R)(R\;R\;B)(R\;R\;R)\)}
- At least one black
- \((E_2)\)={\(B\;B\;B)(B\;R\;B)(B\;B\;R)(B\;B\;R)(R\;B\;B)(R\;B\;R)(R\;R\;B)\)}
- \((E_1\;\cap\;E_2)\)={\(R\;B\;R)(R\;B\;B)(R\;R\;B)\)}

There are 5R and 3B marbles

P\((E_1)\)={P\((R\;B\;B)+P(R\;B\;R)+P(R\;R\;B)+P(R\;R\;R)\)}

=\(\Large\frac{5}{8}\;\times\;\frac{3}{7}\;\times\;\frac{2}{6}\;+\;\frac{5}{8}\;\times\;\frac{3}{7}\;\times\;\frac{4}{6}\;+\;\frac{5}{8}\;\times\;\frac{4}{7}\;\times\;\frac{3}{6}\;+\;\frac{5}{8}\;\times\;\frac{4}{7}\;\times\;\frac{3}{6}\)

We can also do

P(first ball is red)=\(\large\frac{5}{8}\)

P\((E_1)\)

P\((E_2)\)=P(at least 1 black)

=1-P(no black)

=1-p\((R\;R\:R)\)

=1-\(\large\frac{5}{8}\times\)\(\large\frac{4}{7}\times\)\(\large\frac{3}{6}\)

=\(\large\frac{23}{28}\)

=\(\Large\frac{5}{8}\;\times\;\frac{3}{7}\;\times\;\frac{4}{6}\;+\;\frac{5}{8}\;\times\;\frac{3}{7}\;\times\;\frac{2}{6}\;+\;\frac{5}{8}\;\times\;\frac{4}{7}\;\times\;\frac{3}{6}\)

=\(\large\frac{5}{8}\times\)\(\large\frac{5}{7}\)

=\(\Large\frac{25}{56}\)

Requier probability=\(\large\frac{25}{56}\times\)\(\large\frac{8}{5}\)

=\(\Large\frac{5}{7}\)