GTO Brainteaser #10 -- Flipping coins

While this problem isn't directly related to game theory, it does bring up some very interesting concepts from probability which is another essential mathematical topic in poker.  Its also a really great brainteaser so I decided to go ahead and post it even if it is 100% GTO related.  Disclaimer, I didn't not make this problem up, its a very old problem often taught in statistics that I was recently reminded of by reddit.

Problem:

You go to a casino and see that they've introduced a new game.

They are flipping a fair coin and to play you must make a $100 bet.  They will start flipping the coin and keeping track of the outcomes of heads or tails.  They will keep flipping until they either get the sequence heads, heads, tails or heads, tails, tails over their last 3 flips.

If heads, heads, tails comes first the game ends and you lose your bet of $100 for a loss of $100.  If heads, tails, tails comes first the game ends and you win $105 (plus you get your $100 back) for a profit of $105.

So for example if they get:

HTHTHTT

Then you win.  On the other hand if it comes

THTHHT

then you lose.

Is the game positive EV?  What is the EV of the game?

Bonus:  On average, when you win how many flips does this game take?  What about when you lose, what is the average number of flips in that case?

Also, just incase you missed some of my past brainteasers here are links to the past problems (except the true/false quizzes).

• GTO Brain Teaser #1: Exploitation and Counter-Exploitation in Rock Paper Scissors
• GTO Brainteaser #2 -- Red or Black
• GTO Brainteaser #3 -- Blind Man's Bluff
• GTO Brainteaser #5 -- Bayesian Werewolves
• GTO Brainteaser #6 -- Barreling and Bluffing
• GTO Brainteaser #8 -- Solving the Turn, Theory vs Practice
• GTO Brainteaser #9 -- Multistreet Theory: Range Building with Draws