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Question

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


A
62C(15)6
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B
66C(15)6(45)0
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C
(45)6
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D
65C(15)5(45)
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Solution

The correct option is B $$_{ 6 }^{ 6 }{ C }\left (\displaystyle\frac { 1 }{ 5 } \right )^{ 6 }\left (\displaystyle\frac { 4 }{ 5 }\right )^{ 0 }$$
As chance of winning is  $$ \dfrac {1}{5} $$.
For a series of $$6$$ games, the probability that $$A$$ will win all the $$6$$ games, 
$$n= 6$$ and $$k = 6$$ probability of win $$= p = $$
 $$ \dfrac {1}{5} $$ 
Probability for $$k$$ success is  = $$ ^{n}C_{k}\left ( p \right )^{k}\left ( 1-p \right )^{n-k} $$  
Probability $$=$$  $$ ^{6}C_{6}\left ( \dfrac {1}{5} \right )^{6}\left ( 1- \dfrac {1}{5}\right )^{6-6} $$ 
$$ =$$  $$ ^{6}C_{6}\left ( \dfrac {1}{5} \right )^{6}\left ( \dfrac {4}{5}\right )^{0} $$ 

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