I roll a fair die repeatedly until a number larger than 4 is observed. Let N be the total number of times until a number larger than 4 is observed. Find Pr(N=k) for k=1,2,3

A number greater than 4 may be observed with probability ${2\over6}={1\over3}$ on each trial. Therefore, this experiment may be thought of as repeating a Bernoulli experiment with probability of success $p={1\over3}$ until the first success is observed. Therefore, the random variable $N$ has a Geometric distribution with $p={1\over3}$. We can write, $N\sim Geometric ({1\over3})$ given as,

$p(N=k)=({1\over3})({2\over3})^{k-1},\space k=1,2,3,..........\\0, \space otherwise$

Given the value of $k$, the required probability can be evaluated using the probability mass function of the a Geometric random variable given above.