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Do You Dare?

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?