Gambler's Ruin

God is a child; and when he began to play, he cultivated
 mathematics.  It is the most godly of man's games.  -V. Erath
 

Imagine he following gambling scenarios (simplified version of roulette).

 

  1. You play a game where you bet $1 per flip on the flip of a coin winning $1 (plus your bet) if you choose correctly. 

    You choose once heads or tails and play the game over and over.  Assuming you start with $20 and that you are

    successful if you win $20, answer the following questions:

    2.  

    1. How often do you loose your $20?
    2. How often are you successful?
    3. What is the average number of flips necessary to win?
    4. What is the average number of flips necessary to lose?
    5. Answer the same questions above if you quit only after winning $100
    6. Find a relationship between the probability that you are successful and your starting amount and the amount you want to win.

 

 

  1. You are playing roulette. You bet $1 per spin and bet on black or red winning $1 (plus your bet) if you choose correctly. 

    Assume there are 66 spaces on the wheel 32 black, 32 red, and 2 green.  Assume further that as before you start with $20 and that are successful if

    you win $20.  Answer the following questions:

    2.  

    1. How often do you loose your $20?
    2. How often are you successful?
    3. What is the average number of flips necessary to win?
    4. What is the average number of flips necessary to lose?
    5. Answer the same questions above if you quit only after winning $100

f.        Find a relationship between the probability that you are successful and your starting amount and the amount you want to win.

    1. How does this situation differ from the one above?

 

 

3.      For each of the games above program a simulation that plays the game 1000 times and plots the number of times you played vs. how much you lose. 

Fix the axis so that x varies from (0, 1000) and y varies from (-100, 100) In this situation assume you have infinitely many dollars and that you never quit. 

Do this with other games having higher and lower probabilities of  winning.

 

a.       Describe what is happening?

b.      Make a conjecture about the relationship of the slope of the “line” vs. the probability of a win.