at around 20 mins I think you make a mistake. It is a technicality, but E [T_3 | X_2 =1 ] = pD + (1-p) 0 + E[T_2] , and so simply adding your 2 eqns together gives what you need. and then you dont have to add in the E[T_2] as that is baked into the conditional expectation where X_2 = 1. I think the mistake happens because you are treating T_n as two types of variable -- both cumulative payout, as well incremental pay out on toss number n. imagine you would have to introduce some third variable for incremental payouts to avoid this mistake
