Finding probability using moment-generating functions

Note that the distribution of the random variable $Y$ which is $0$ with probability $0.2$ and $1$ with probability $0.8$ has the same mgf: to check, compute. Now use uniqueness.

Like in the case of the Laplace Transform, we often recognize an mgf as being a familiar one, and thereby identify a distribution.

Detail: Let $Y$ be a Bernoulli random variable which is $0$ with probability $a$, and $1$ with probability $b$. Then $$E(e^{tY})=a e^{(0)t}+be^{(1)(t)}=a+be^t.$$ Conversely, by the uniqueness theorem, a random variable whose distribution has mgf $\,a+be^t$ is a Bernoulli random variable, taking on the value $0$ with probability $a$, and $1$ with probability $b$.

We have found that our random variable $X$ has distribution with mgf $\,0.2+0.8e^t$. Therefore $X$ must be a Bernoulli random variable, with $\Pr(X=0)=0.2$ and $\Pr(X=1)=0.8$.