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Consider a sample of size $3$drawn in the following manner: We start with an urn containing $5$white and $7$red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color.

Condition is$0$white ball.

R is red ball

W is white ball

The solution is given below,

$0\text{white}鈬�(R,R,R)$

$\mathrm{p}=\frac{7}{12}脳\frac{8}{13}脳\frac{9}{14}$

$=\frac{3}{13}$

The probability that the sample will contain exactly$0$white ball is$\frac{3}{13}$.

Consider a sample of size $3$drawn in the following manner: We start with an urn containing $5$white and $7$red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color.

Condition is $1$white ball.

R is red ball

W is white ball

The solution is given below,

$1\text{White}鈬�(\mathrm{W},\mathrm{R},\mathrm{R})+(\mathrm{R},\mathrm{W},\mathrm{R})+(\mathrm{R},\mathrm{R},\mathrm{W})$

$p=\frac{5}{12}脳\frac{7}{13}脳\frac{8}{14}+\frac{7}{12}脳\frac{5}{13}脳\frac{8}{14}+\frac{7}{12}脳\frac{8}{13}脳\frac{5}{14}$

$=3脳\frac{5}{12}脳\frac{7}{13}脳\frac{8}{14}$

$=\frac{5}{13}$

The probability that the sample will contain exactly$1$white ball is$\frac{5}{13}$.

Consider a sample of size $3$drawn in the following manner: We start with an urn containing $5$white and $7$red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color.

Condition is $3$white balls.

R is red ball

W is white ball

The solution is given below,

$3\text{White}鈬�(\mathrm{W},\mathrm{W},\mathrm{W})$

$\mathrm{p}=\frac{5}{12}脳\frac{6}{13}脳\frac{7}{14}$

$=\frac{5}{52}$

The probability that the sample will contain exactly$3$white ball is$\frac{5}{52}$.

Consider a sample of size $3$drawn in the following manner: We start with an urn containing $5$white and $7$red balls. At each stage, a ball is drawn and its color is noted. The ball is then returned to the urn, along with an additional ball of the same color.

Condition is $2$white balls

R is red ball

W is white ball

The solution is given below,

$2$white balls

$(\mathrm{W},\mathrm{W},\mathrm{R})+(\mathrm{W},\mathrm{R},\mathrm{W})+(R,W,W)$

$\mathrm{p}=\frac{5}{12}脳\frac{6}{13}脳\frac{7}{14}+\frac{5}{12}脳\frac{7}{13}脳\frac{6}{14}+\frac{7}{12}脳\frac{5}{13}脳\frac{6}{14}$

$=3脳\frac{5}{12}脳\frac{6}{13}脳\frac{7}{14}$

$=\frac{15}{52}$

The probability that the sample will contain exactly $2$ white ball is $\frac{15}{52}$.