Question

I roll a fair die repeatedly until a number larger than 4 is observed. If N is the total number of times that I roll the die, find P(N = k), for k = 1,2,3,....

Answer 1

The total number of outcomes in each trial is 6, the number of favorable outcomes is 2 (5 and 6 are larger than 4).

Probability of observing a number larger than 4 on a single die roll is $\cfrac{2} {6} =\cfrac{1}{3},$ probability of ob observing a number less than or equal to 4 is $\cfrac{2} {3}.$

$N=k$ means that $k-1$ times we had observed a number less than or equal to 4 before we got 5 or 6.

All the rolls are independent events,

$P(N=k) =\begin{pmatrix} \cfrac{2} {3} \end{pmatrix}^{k-1} \cdot\cfrac{1}{3}.$