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Question
A fair coin is tossed 100 times. The probability of getting tails an odd number of times is
Options
1/2
1/8
3/8
None of these
Solution
1/2
Here n=100
Let X denote the number of times a tail is obtained.
\[\text{ Here } , p = q = \frac{1}{2}\]
\[P(X = \text{ odd} ) = P(X = 1, 3, 5, . . . . 99) \]
\[ = \left( ^{100}{}{C}_1 + ^{100}{}{C}_3 + . . . . . + ^{100}{}{C}_{99} \right) \left( \frac{1}{2} \right)^{100} \]
\[ = \text{ Sum of odd coefficients in binomial expansion in}\ (1 + x )^{100} \left( \frac{1}{2} \right)^{100} \]
\[ = \frac{2^{(100 - 1)}}{2^{100}}\]
\[ = \frac{1}{2}\]
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