Question-43 A fair coin is tossed 100 tirnes. The probability of getting tails an odd number of times is
A) 1/2
B) 1/4
C) 1/8
D) 3/8
Answer: A) 1/2
Explanation
Probability of getting a tail in a single toss
P =1/2 and not getting tail, q = 1/2
Using Binomial distribution,
\therefore Required probability
\begin{aligned} & =P(X=1)+P(X=3)+P(X=5)+\ldots+P(X=99) \\ & ={ }^{100} C_{1}\left(\frac{1}{2}\right)^{1}\left(\frac{1}{2}\right)^{99}+{ }^{100} C_{3}\left(\frac{1}{2}\right)^{3}\left(\frac{1}{2}\right)^{97}+{ }^{100} C_{5}\left(\frac{1}{2}\right)^{5}\left(\frac{1}{2}\right)^{95}+\ldots+{ }^{100} C_{99}\left(\frac{1}{2}\right)^{99}\left(\frac{1}{2}\right) \\ & =\left(\frac{1}{2}\right)^{100}\left({ }^{100} C_{1}+{ }^{100} C_{3}+{ }^{100} C_{5}+\ldots+{ }^{100} C_{99}\right) \\ & =\frac{1}{2^{100}}\left(2^{99}\right) \\ & =\frac{1}{2} \end{aligned}