Rolling until a six under three conditions

Dan keeps rolling a fair die and stops as soon as a 6 comes up. Find the expected number of rolls in each case.

(a) You are told that no 5 appeared during his rolls.

(b) You are told that at least one 5 appeared.

(c) You are told that every roll he made showed an even number.

(a) $3$; (b) $9$; (c) $\tfrac32$. (a) No 5 means the first roll from $\{5,6\}$ was a 6; the waiting time for that first roll is geometric with $p=\tfrac13$ and independent of which face it is, so the mean is 3. (b) Unconditionally the mean is 6 $=\tfrac12\cdot 3+\tfrac12\cdot E_b$, so $E_b=9$. (c) Likewise the first odd-or-6 roll is geometric with $p=\tfrac23$, so the mean is $\tfrac32$.