This problem looks at the probability of a tie in hacking competitions. Two hackers are
about to compete, each using their proprietary hacking software, to break into a secure
site.
a) They decided to first play a warm up friendly game of pure luck: they simultaneously
toss fair coins to see who will toss the first \"heads\". What is the probability that the
outcome is a tie, i.e., both toss the first \"heads\" at the same time?
b) And next comes the real hacking competition of breaking into that secure site. Let
discrete random variables $X$ and $Y$ be the number of hacking attempts until success
by hackers $H_x$ and $H_y$, respectively. It turns out that $X$ and $Y$ are geometric random
variables with parameters $p$ and $q$, respectively. Thus the PMF of $X$ is
$P_X(x) = \begin{cases} p(1-p)^{x-1} & x = 1,2,...\\0 & \text{otherwise},\end{cases}$
(and similarly for $Y$, using parameter $q$). What is the probability of a tie ($X = Y$)?
