Gregmat math problem - probability

“In a certain game of chance, if a person correctly guesses the side a fair coin lands on, he or she wins $20. If he or she guesses incorrectly, however, he or she must pay $10 of their own money. If a person plays this game five times in a row (five coin flips), what is the probability he or she will lose money?”

i have a problem with this question, if it says that the person looses money when she does not guess correctly the coin side, this means that we could have 2 cases: she guessed head and it turns out tail and viceversa. now this means we should consider 2 times (1/2)^5 times (the cases in which we have one guess + the case in which we have 0 guesses).

we have 4 cases total:

  1. HTTTT with 5 arrangements; 2. THHHH with 5 arrangements;
  2. TTTTT with only 1; 4. HHHHH with only 1;

If you’re going that route, then the sample space should have 4^5 elements not 2^5