Original problem

Sue and Bob take turns rolling a 6-sided die. Once either person rolls a 6 the game is over. Sue rolls first, if she doesn't roll a 6, Bob rolls the die, if he doesn't roll a 6, Sue rolls again. They continue taking turns until one of them rolls a 6.

Bob rolls a 6 before Sue.

What is the probability Bob rolled the 6 on his second turn?

First, an easy thing.

Probability Bob rolls a 6 on turn 2 given nothing at all.

Sue rolls 1-5.
Bob rolls 1-5.
Sue rolls 1-5.
Bob rolls the 6.

So the probability of it happening at all is

But what is the probability given that Bob rolls the first 6?

Wrong Answer

but I'm not sure what's invalid about it.

Sue rolls 1-5.
Bob rolls 1-5.
Sue rolls 1-5.
Bob rolls the 6.

What is the probability that Bob rolls the first 6?

Probability of getting a 6 on his first turn
Probability of getting a 6 on his second turn
Probability of getting a 6 on his th turn
Probability of rolling the first 6

Conditional probability woo

So now we can find the probability he won on turn 2 given that he won.

Final Result

. See? It's easy. :/