91Ó°ÊÓ

The box contains r red balls and w white balls.

Suppose that balls are drawn from the box one at a time, at random, without replacement.

We have, r red balls and w white balls.

So, the total numbers of balls are\(r + w\).

We can choose r number of red balls from total number of balls\(r + w\)by,\({}^{r + w}{C_r}\)ways.

a.

Since there are\({}^{r + w}{C_r}\)total ways of drawing the balls, only these have the red balls first.

So, the probability of drawing all the red balls before any white ball is,\(\frac{1}{{{}^{r + w}{C_r}}}\)

b.

If we want to draw all red balls before two white balls are drawn, then all red balls must be in first\(r + 1\)draws.

There will be exactly one white ball in the first\(r + 1\)draws.

So, the probability of drawing all red balls before two whites’ balls is, \(\frac{{r + 1}}{{{}^{r + w}{C_r}}}\)