Question

In a certain game $A^{\prime }s$ skill is to $B^{\prime }s$ as $3$ to $2:$ find the chance of $A$ winning $3$ games at least out of $5.$

Answer 1

Probability of A winning the game $=\frac{3}{5}$

Probability of A losing the game $=\frac{2}{5}$

We need to find the probability of A winning either 3 or 4 or 5 games

Probability of A winning 5 games $={\left(\frac{3}{5}\right)}^5$

Probability of A winning 4 games and losing 1 game $=5\cdot {\left(\frac{3}{5}\right)}^4\cdot \left(\frac{2}{5}\right)$

Probability of A winning 3 games and losing 2 game $=10\cdot {\left(\frac{3}{5}\right)}^3\cdot {\left(\frac{2}{5}\right)}^2$

$\therefore$ Required Probability $={\left(\frac{3}{5}\right)}^5+5\cdot {\left(\frac{3}{5}\right)}^4\cdot \left(\frac{2}{5}\right)+10\cdot {\left(\frac{3}{5}\right)}^3\cdot {\left(\frac{2}{5}\right)}^2=\frac{2133}{3125}$