Question

The probability that Dhoni will hit century in every ODI matches he plays is $\frac{1}{5}$. When he plays $6$ matches in World Cup 2011, the probability that he will score at least one century is:

• A. $\frac{2357}{3125}$
• B. $\frac{121}{3125}$
• C. $\frac{768}{3125}$
• D. $\frac{4096}{15625}$

Answer 1

The correct option is D

$\frac{4096}{15625}$

Probability of getting a century, $p=\frac{1}{5}$

Probability of not getting a century, $q=\frac{4}{5}$

$\Rightarrow$ Total no. of match $=6$

By Bernoulli's equation,

$\Rightarrow p={}^n{C}_r{p}^r{q}^{n-r}$

p ( getting atleast $1$ ) $={}^6{C}_1\left(\frac{1}{5}\right){\left(\frac{4}{5}\right)}^5+{}^6{C}_2{\left(\frac{1}{5}\right)}^2{\left(\frac{4}{5}\right)}^4+{}^6{C}_3{\left(\frac{1}{5}\right)}^3{\left(\frac{4}{5}\right)}^3+{}^6{C}_4{\left(\frac{1}{5}\right)}^4{\left(\frac{4}{5}\right)}^2+{}^6{C}_5{\left(\frac{1}{5}\right)}^5\left(\frac{4}{5}\right)+{}^6{C}_6{\left(\frac{1}{5}\right)}^6$

$=\frac{6\times 1024+15\times 256+20\times 64+15\times 16+6\times 4+1}{5^6}$

$=\frac{11529}{15625}$

$\Rightarrow p=1-\frac{11529}{15625}$

$=\frac{4096}{15625}$

Hence, the answer is $\frac{4096}{15625}.$