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  • A fair coin is tossed times. The probability that the difference between the number of heads and tail

A fair coin is tossed \mathrm{m} times. The probability that the difference between the number of heads and tails in (n-3) is

Option: 1

2^{-n}


Option: 2

0


Option: 3

{ }^{n} \mathrm{C}_{n-3} 2^{2-n}


Option: 4

2^{-n+3}


Answers (1)

best_answer

Let no. of heads = x
no. of tails = y

Given \mathrm{x+y =n}
\mathrm{|x-y| =n-3}
\mathrm{|x-(n-x)| =n-3}
\mathrm{|2 x-n| =n-3}
\mathrm{\Rightarrow \quad(2 x-n) = \pm(n-3)}

\mathrm{\text { case 1: } }
\mathrm{\Rightarrow 2 x-n =n-3 }
\mathrm{2 x =2 n-3 }
\mathrm{x =\frac{2 n-3}{2} }
(not possible, as \mathrm{x } and \mathrm{n } are natural numbers)

\mathrm{Case\, II: }
\mathrm{2 x-n=-n+3 }
\mathrm{\Rightarrow \quad x=\frac{3}{2} \text { (Not possible) } }

\mathrm{\therefore }  Required probability \mathrm{= 0 }


          

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Sayak

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