Q256 / 270 · NDA 2016 (Phase 1) · Mathematics

A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?

  • Correct answer
    A
    $\frac{1}{2}$
  • B
    $\frac{3}{8}$
  • C
    $\frac{1}{4}$
  • D
    $\frac{1}{8}$

Solution

Let x denote number of tails. Then, X is a binomial variate with parameters: $\begin{aligned} & \mathrm{x}=100 \quad \& \quad \mathrm{p}=\frac{1}{2} \\ & \therefore \mathrm{p}(\mathrm{x}=\mathrm{r})={ }^{100} \mathrm{C}_{\mathrm{r}}\left(\frac{1}{2}\right)^{100} ;(\mathrm{r}=0,1,2, \ldots \ldots \ldots . .100) \\ & \text { Req. probability }=\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=3)+\ldots .+\mathrm{P}(\mathrm{x}=99) \\ & = \\ & {\left[\frac{1}{2}\right]^{100}\left[{ }^{100} \mathrm{C}_1+{ }^{100} \mathrm{C}_3+\ldots .+{ }^{100} \mathrm{C}_{99}\right]} \\ & \quad=\frac{1}{2^{100}} 2^{99}=\frac{1}{2} \end{aligned}$

Hence, the correct answer is option: 'A'.

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