Question

From a bag containing $4$ white and $5$ black balls a man draws $3$ at random$;$ what are the odds against these being all black?

Answer 1

The total number of ways in which $3$ balls can be drawn is ${}^9{C}_3$, and the number of ways of drawing $3$ black balls is ${}^5{C}_3;$ therefore the chance of drawing $3$ black balls
$=\frac{{}^5{C}_3}{{}^9{C}_3}=\frac{5\cdot 4\cdot 3}{9\cdot 8\cdot 7}=\frac{5}{42}$.
Thus the odds against the event are $37$ to $5$.