Question

If an unbaised coin is tossed n times. Find the probability that head appears an odd number of times.

Answer 1

$\because$ Probability of getting a head in a single trial $=\frac{1}{2}$
and probability of getting a tail in a single trail $=\frac{1}{2}$
Now, $P(X=odd)=P(X=1)+P(X=3)+P(C=5)+...$
${=}^n{C}_1{\left(\frac{1}{2}\right)}^n{+}^n{C}_3{\left(\frac{1}{2}\right)}^n{+}^n{C}_5{\left(\frac{1}{2}\right)}^n+...$
$={\left(\frac{1}{2}\right)}^n({}^n{C}_1{+}^n{C}_3{+}^n{C}_5+...)$
$=\frac{1}{2^n}.2^{n-1}=\frac{1}{2}$