A coin is tossed $3$ times. If $X$ is a random variable representing the number of tails, then the mathematical expectation of $X$ is:

This is a Binomial Distribution problem with $n=3$ and $p=P(\text{Tail})=1/2$.The expectation $E(X)$ for a binomial distribution is $np$:
$$E(X)=3\times \frac{1}{2}=\frac{3}{2}$$
Alternatively, possible values of $X$ are $0,1,2,3$ with probabilities:$P(0)=1/8$, $P(1)=3/8$, $P(2)=3/8$, $P(3)=1/8$.$E(X)=\sum x_i{p}_i=0(1/8)+1(3/8)+2(3/8)+3(1/8)=(3+6+3)/8=12/8=3/2$.