Question

[10 pts.] You keep rolling a fair 6-sided die as long as no value is repeated. When you see the first repeated value, that is your last roll. Let X be the number of rolls it took. Find P(X = k) for all k.

You must justify every single step to get to the answer, or no credit will be awarded.

Answer 1

Let's say the value gets repeated after rolling twice, P(2) =

Similarly, P(3) = ( The first value has to be different from the last two , both of which are same. i.e Lets say the last two values are 1, 1st roll has to be one of the 2,3,4,5,6)

P(4) = (The 2nd roll has to be different from the last two , both of which are same. i.e Lets say the last two values are 1, 2nd roll has to be one of the 2,3,4,5,6 . Lets say it is 2.. Now, 1st roll has to be one of the 1,3,4,5,6 as 2 cant be repeated again & it can all the other values.)

So, to generalize , P(k) =   for k>0

Hope I clarified your query. Do let me know, your thoughts on this.