We are given two coins, the first shows head with probability $p∈(0,1)$ while the second with probability $q∈(0,1)$. We toss these coins simultaneously and, if both show head, we stop. If not, we toss them again. What is the probability that we need to toss the two coins at most k times before obtaining two heads?
I am unsure here of what distribution to use to solve this, and the parameters involved. At first I was thinking binomial distribution because it seems to fit the requirements for it, but I'm not entirely sure. Any advice would be greatly appreciated.
The distribution you are facing is a geometric
$$X\sim Geo(pq)$$
defined on $x=1,2,3...$
Thus
$$P(X\leq k)=1-(1-pq)^k$$