This problem was found in the discussion section of the resource Math2.org that was reviewed this week. I have taken the liberty to make a few alterations.
You start with thirteen cards labelled 1 to 13.
They are randomly shuffled (7 times!) and then arranged face-down in front of you.
You pay the house $1 to pick a card at random.
If the selected card is below 7, you win $1 (you get $2 back), otherwise the house keeps your $1.
At the start, you have $10.
What is the probability that you will reach $15 before you lose all of your $10?
Hint: The probability of losing your $1 is 7/13, while the probability of winning $1 is 6/13.
Simulate several rounds of the game to get a feel for what is happening...
Can you draw a tree diagram to represent the situation...and perhaps this leads to a some equations to solve?
How can this be turned into a single probability?
Solution Commentary: Rather than provide my own commentary, why not browse the original thread in the discussion surrounding this problem.
Do you agree with the mathematics being done?