Starting from scratch, they first need to get a head. This takes 2 tosses on average (1 with 50% probability, 2 with 25% probability, 3 with 12.5% probability, etc.). At that point, both Alice and Bob have 50% chance of getting the target sequence with one additional toss.
In the case of failure, Alice still has 50% chance of success in each subsequent toss. On average she will need two additional tosses to get a tail and the answer is 2+2=4.
In the case of failure, Bob has to start again. If we call the answer x, we can write x=2+0.5 1+0.5 (1+x) and solving the equation we get x=6.
In the case of failure, Alice still has 50% chance of success in each subsequent toss. On average she will need two additional tosses to get a tail and the answer is 2+2=4.
In the case of failure, Bob has to start again. If we call the answer x, we can write x=2+0.5 1+0.5 (1+x) and solving the equation we get x=6.