Interpretation of probability computed with mixture model Suppose I have a mixture model: 
$$
p(x) = \sum_{j=1}^K\lambda_j\phi(x;\theta_j),\quad \lambda_j\geq 0,\sum_{j=1}^K\lambda_j = 1
$$  I understand what it means to sample a random variable $X\sim p$ - first you select $J\sim \text{Cat}(\boldsymbol{\lambda})$, then draw $X\vert j\sim \phi(x,\theta_j)$.  However, I'm interested in hearing an elementary interpretation of probabilities computed with $p$, i.e. suppose I compute 
$$
\Bbb{P}[X\in A] = \int_Ap(x)dx = \sum_{j=1}^K \lambda_jP_j,\quad P_j = \int_A\phi(x;\theta_j)dx
$$  How would I describe what this probability means to a complete beginner/non-statistician?  
 A: I find it hard to come up with a general explanation that is both meaningful and understandable to everybody. I decided to go with an answer that is understandable, also to complete outsiders, sacrificing the meaningful part for some. The downside of this is that some (many) things will be oversimplified and even theoretically inaccurate. However, I often find it helpful to start imprecisely to then continuously fill in details later on. 
I'll go with with a simple leading example of the labour force and bit-by-bit generalise the different parts of the example.
You could start by saying there are $K$ groups of workers in a market, altogether the groups of workers form the labour market. Every group of workers is different. Also, some groups of workers may be large and some may be small. The relative sizes of these groups are known ($\lambda_k$ for group $k$). 
Suppose we know from every group what their salary distribution looks like, namely a function called $\phi(x;k)$ for group $k$. This function gives me the probability that the salary of a worker in group $k$ has salary $x$. If we want to know the probability that a worker in group $k$ has salary a between 30k and 50k, we'd calculate $\int_{30k}^{50k}\phi(x;k)dx$. (See Wikipedia: Cumulative Distribution Function for more info)
Now suppose we want to know the probability that the income of a completely randomly chosen worker lies between \$30k and \$50k, that is, $P(X \in [30k - 50k])$ (or more generally $P(X \in A)$ for some set $A$). It makes sense to view the picking of a random person from the labour force as a two-stage process:


*

*Pick one of the $K$ groups

*Find the probability of a salary in $[30k-50k]$ for this group


We can write the above two steps in terms of probabilities:


*

*The probability of picking group $k$ is equal to $\lambda_k$. 

*The probability of a salary in $[30k-50k]$ in group $k$ is equal to $\int_{30k}^{50k}\phi(x;k)dx$


If we repeat step 1 and 2 for all groups $1,2,\dots,K$, we get the probability of a salary in $[30k-50k]$ for the entire population:
\begin{equation}
\sum_{k=1}^{K} \lambda_k \cdot \int_{30k}^{50k}\phi(x;k)dx
\end{equation}
This can be viewed as a weighted average of the probability of a salary between 30k and 50k in every workers group. The weights are given by the relative size of that workers group.
