Consider a coin that comes up heads with probability $1/3$ and, thus, tails with probability

$2/3.$ Consider the following recursive algorithm Heads, which takes as input a positive

integer k:

Algorithm Heads$($k$)$:

$//$ all coin flips made are mutually independent

flip the coin;

if the coin came up heads

then return k $+1$

else Heads$($k $+1)$

endif

You run algorithm Heads$(1),$ i$.$e$.,$ with k $=1.$ Define the random variable X to be the value

of the output of this algorithm. Let m $>=1$ be an integer. What is Pr$($X $=$ m $+1)?$

$($a$)2$m$1/3$

m

$($b$)(2/3)$m

$($c$)2$m$/3$

m$+1$

$($d$)(2/3)$m$+1$