An unbiased die is successively rolled. Let $X$ and $Y$ denote, respectively, the number of rolls necessary to obtain a six and a five. $E[X]= 6$.
find $E[X \mid Y=1]$
Iam stuck on this. Iam thinking that if it requires one throw to get a 'five' then if i start throwing then the first throw is a 'five',since it was not a 'six' i need to keep on throwing ,but then i already threw the dice once therefore: $E[X \mid Y=1] = E[X] -1 $
but wrong according to my book so can someone explain whats wrong with my reasoning
If the first trial yielded a $5$, then you're starting with the second trial in your attempt to get a $6$. So it's $1$ plus the average number of trials needed to get a $6$, i.e. it's $1+6$.