We basically have four cases for Bob:
- HH: terminates for Bob.
- HT: Bob restarts the sequence.
- TT: Bob restarts the sequence.
- TH: This is a continuation. It degenerates into the answer to a single throw. Otherwise its just a recursion.
Then we have another four cases for Alice:
- HH: This is a continuation. It degenerates into the answer to a single throw or else a recursion.
- HT: terminates for Alice.
- TT: Alice restarts the sequence.
- TH: This is a continuation. It degenerates into the answer to a single throw. Otherwise its just a recursion.
So with that understanding it is a bit more clear how we get this result:
Alice has one win and two opportunities for a win and one for a restart. Bob has one win and one opportunity for a win and two restarts.
That was a bit confusing. I wonder how the problem could be worded to ensure that people got the answer correctly every time?
Multiple heads in a row are more bunched than transition sequences because, for example, a sequence of three heads in a row will include two sequences with two heads in a row. You can't do that with a transition sequence--it takes at least four tosses to get two identical transition sequences.