Question

A box contains $20$ identical balls of which $5$ are white and $15$ black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn fro the $3^{rd}$ time on $6^{th}$ drawn is

Answer 1

We have,

Probability of getting white ball is $=\frac{5}{20}=\frac{1}{4}$

Now,

Probability of given white balls is drawn $3^{rd}$ times on $6^{th}$ drawn =1.

Getting 2 white balls in 5 drawn $\times$ Probability of white balls in $6^{th}$ terms.

$=\left({}^5{C}_2\right)\times {\left(\frac{5}{20}\right)}^5\times \frac{5}{20}$

$=\frac{5!}{3!\times 2!}\times \frac{5}{20}\times \frac{5}{20}\times \frac{5}{20}\times \frac{5}{20}\times \frac{5}{20}\times \frac{5}{20}\times \frac{5}{20}$

$=\frac{5\times 4}{2\times 1}\times \frac{15625}{64000000}$

$=\frac{1562510}{64000000}$

$=\frac{1}{2048}$

Hence, this is the answer.