Mean gap between thank-you cards from two firms

Clients arrive at two quant firms according to independent Poisson processes with rates $6$ and $10$. Each client of the first firm leaves a glowing review with probability $\tfrac16$, and each client of the second with probability $\tfrac15$, independently. Every client who leaves a glowing review is mailed a thank-you card. What is the expected waiting time between successive card-receiving clients (across both firms)?

$1/3$. Thinning a Poisson process keeps it Poisson: the card streams have rates $6\cdot\tfrac16=1$ and $10\cdot\tfrac15=2$. Superposing independent Poisson streams adds rates, giving rate $3$, so gaps are $\text{Exp}(3)$ with mean $\tfrac13$.