Question

$A$ and $B$ play a game in which $A^{{}^{\prime }}s$ chance of winning is $\frac{1}{5}$ in a series of $6$ games, the probability that $A$ will win all the $6$ games is

• A. ${}_2^{6}C{\left(\frac{1}{5}\right)}^6$
• B. ${}_6^{6}C{\left(\frac{1}{5}\right)}^6{\left(\frac{4}{5}\right)}^0$
• C. ${\left(\frac{4}{5}\right)}^6$
• D. ${}_5^{6}C{\left(\frac{1}{5}\right)}^5\left(\frac{4}{5}\right)$

Answer 1

The correct option is B ${}_6^{6}C{\left(\frac{1}{5}\right)}^6{\left(\frac{4}{5}\right)}^0$

As chance of winning is $\frac{1}{5}$.
For a series of $6$ games, the probability that $A$ will win all the $6$ games,
$n=6$ and $k=6$ probability of win $=p=$ $\frac{1}{5}$

Probability for $k$ success is = ${}^n{C}_k{\left(p\right)}^k{(1-p)}^{n-k}$

Probability $=$ ${}^6{C}_6{\left(\frac{1}{5}\right)}^6{(1-\frac{1}{5})}^{6-6}$

$=$ ${}^6{C}_6{\left(\frac{1}{5}\right)}^6{\left(\frac{4}{5}\right)}^0$