Survival probability of a mixture of exponential and gamma lifetimes

A mobile manufacturing company uses two brands of batteries for its mobiles. The life (in years) of batteries of Brand I follows an exponential distribution with the probability density function

$$f(x)=\begin{cases}e^{-x},&x>0,\\0,&\text{otherwise,}\end{cases}$$

and that of Brand II follows a gamma distribution with the probability density function

$$g(x)=\begin{cases}\dfrac x4e^{-x/2},&x>0,\\0,&\text{otherwise.}\end{cases}$$

The company uses the batteries of Brands I and II in proportion of $20\%$ and $80\%$ respectively, in its mobiles. The probability that a randomly selected mobile has the battery life more than $2$ years is