A person earns $x_i$ amount of money every month where $x_i$ is an exponential random variable with parameter $\lambda_1$. The amount $x_i(1-p)y$, here $0 \leq p\leq 1$ and $y$ is exponentially distributed with parameter $\lambda_2$, is invested every month in lottery $L_1$ while amount $x_ip$ is saved every month. As soon the saved amount becomes at least $Q$ then in that month $Q\cdot z$ is invested in lottery $L_2$ and no money is invested in lottery $L_1$. Here $z$ is exponentially distributed with parameter $\lambda_3$.
The person wins the lottery $L_1$ if $x_i(1-p)y>J$ and he wins lottery $L_2$ if $Qz>K$ in any month. All the $J, K, Q$ are constants. The person keeps on investing the money in the above fashion irrespective of winning or losing either of the lottery. I need to find the probability that the person does not win lottery $L_1$. I also need to find the probability that the person does not win lottery $L_2$.
To this end I have formulated the probabilities \begin{align*} Pr(\text{person lose }L_1) &= Pr(x_i(1-p)y<J\mid \text{no investment is made in } L_2) \\ &\phantom{=~} \cdot Pr(\text{no investment is made in }L_2)+Pr(\text{investment is made in } L_2) \end{align*} and \begin{align*} Pr(\text{person lose } L_2) &= Pr(\text{no investment is made in } L_2) \\ &\phantom{=~} +Pr(Qz<K \mid \text{investment is made in } L_2) Pr(\text{investment is made in } L_2). \end{align*}
I want to know if these are right expressions or not. If they are right then is there a way to find $Pr(\text{investment is made in } L_2)$? I will be very thankful for any help in this regard. I will appreciate if somebody could point me to a reference where I can find a similar problem.
I assumed that the result of both lotteries are not independent because the investment in lottery $L_1$ depends on the investment in $L_2$. In particular if investment is made in $L_2$ in any particular month then in that month no investment is made in $L_1$.
Another form of $Pr(\text{person lose } L_2)$:
After some brain storming I have some reasoning to solve the $Pr(\text{person lose } L_2)$. As it can be observed the money saved for lottery $L_2$ is a chain process. Hence, if we could find some steady state distribution for the amount of money left after every time the person invest in $L_2$ then we can write the $Pr(\text{person lose } L_2)$ as follows \begin{align*} Pr(\text{person lose } L_2)=\sum_{p=1}^{\infty}Pr(Qz<K|\text{investment is made after p months})Pr(\text{investment is made after p months}) \end{align*} However this gives the answer \begin{align*} Pr(\text{person lose } L_2)=Pr(Qz<K) \end{align*} I am not sure this reasoning is right or wrong. any help will be much appreciated.