Question
A fair die is to be rolled repeatedly until a six comes up. Find the median of the total number of rolls given that five comes up on the first roll.
My working
Clearly, this follows a geometric distribution with $p = \frac 1 6$ and the median is given by $$-\frac 1 {log_2(1 - p)}$$ which evaluates to $4$, rounded up.
However, how do we factor in the fact that five comes up on the first roll? Any intuitive explanations will be greatly appreciated!
The die is memoryless, meaning that it doesn't "remember" how many times it has failed to roll a $6$ so far. That means that the distribution for the number of rolls, conditioned on not rolling a $6$ on the first roll, is simply the original distribution of number of rolls, shifted to the right by $1$.
Symbolically, if we denote by $p_k$ the probability of rolling $k$ times, irrespective of the first roll, then the probability $q_k$ of rolling $k$ times, given that the first roll was not a $6$, is given by $q_1 = 0$ and $ q_k = p_{k-1}, k \geq 2$.
As a result, the median—as well as the mean, for that matter—is just shifted over by $1$ as well. So if you computed the median to be $4$ before, then now the new information just makes it $5$. (I'm not sure how median is defined for this problem.) Similarly, the original mean was $6$, so the new mean is $7$.