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Q.
If a fair coin is tossed 20 times and let we get head n times, then probability that 21 is odd, is
Rajasthan PETRajasthan PET 2009
Solution:
Probability of getting head in one trial,
$ p=1/2 $ and probability of not getting head,
$ q=1/2 $ Probability of getting head odd times
$ {{=}^{20}}{{C}_{1}}{{\left( \frac{1}{2} \right)}^{1}}{{\left( \frac{1}{2} \right)}^{19}}{{+}^{20}}{{C}_{3}}{{\left( \frac{1}{2} \right)}^{3}}{{\left( \frac{1}{2} \right)}^{17}} $
$ +....{{+}^{20}}{{C}_{19}}{{\left( \frac{1}{2} \right)}^{19}}{{\left( \frac{1}{2} \right)}^{1}} $
$ =\frac{1}{{{2}^{20}}}{{[}^{20}}{{C}_{1}}{{+}^{20}}{{C}_{3}}+...{{+}^{20}}{{C}_{19}}] $
$ =\frac{1}{{{2}^{20}}}\times {{2}^{20-1}}\frac{{{2}^{19}}}{{{2}^{20}}}=\frac{1}{2} $