Expected earnings of the house per play?

#1
For a project in my class we had to create a casino themed game to earn profit. I settled on this relatively simple game; within a deck of cards (excluding the joker) the number 7 is in the middle. There are 6 cards below and above it. The idea behind the game is to first display a 7 card, and the players try to predict whether the next card in the stack is going to be higher or lower than the 7 shown. There will be no 7s in the stack, so ties cannot be possible. Also we will continuously replace cards so the stack will always have a 50/50 shot of being higher or lower. The most a player can predict correctly in a row before we kick them out is 7 (because we need to profit). THERE IS A FIXED ENTRANCE FEE OF $5. The payout multipliers are shown below:

7x in row : 49x payout

6x: 28x payout

5x: 14x payout

4x: 7x payout

3x: 3x payout

2x: money back

1 and 0x: lose all money

With each correct pick, the player can either choose to bail with their earnings (and obviously keep them) or risk continuation. IF THE PLAYER GUESSES INCORRECTLY, NO MATTER HOW MUCH THEY HAVE GUESSED CORRECTLY IN A ROW BEFORE, THEY LOSE ALL THEIR MONEY. So even one who gets 6 in a row can lose their money if they get the 7th wrong.

The question is, what is the expected earnings of the game hosts (us) for each $5 wager? This may not seem too difficult to some, but I just haven't been able to figure it out. Thanks for the help!
 

Dason

Ambassador to the humans
#2
I think it would be dependent entirely on the strategy the player utilizes. For example if the strategy was only guess once and then quit no matter what - they would be guaranteed to lose every time.

So you need to figure out what strategy you're considering before you can try to assess any possible attributes it would have.