Here is an interesting game presented and discussed by psychologists, mathematicians, philosophers, and people in business, risk management, and gambling. You start with $1, and a die is rolled. You "win" if the outcome is not a 6, in which case you double your amount. You "lose" if the outcome is a 6, in which you lose everything you have accumulated. At any point of this game, you can continue or quit. Once you lose your money, there is no point in continuing since twice 0 is still 0. For example, on the first roll, say a 3 comes up. Now you have $2. You decide to continue, and the second outcome is a 5. Now you have $4. You decide to continue, and the third outcome is a 4. Now you will have $8. You decide to continue, and the four outcome is a 3. You will now have $16. You decide to continue, and the fifth outcome is a 6. You lose everything (your $16) and go down to $0.

(a) If no 6 comes up the first 5 rolls, how much would you have?

(b) Suppose no 6 comes up the first 10 rolls. How much would you have?

(c) If the first 6 comes up on the th roll, how much would you lose then?

(d) There is about a 93.5% chance that at least one 6 comes up during the first 15 rolls. If you have played this game 15 times, therefore, you have about a 93.5% chance that you have nothing, but you still have a 6.5% chance (( 5/6 ) 15 0.0649) that you have money. How much would you have if no 6 comes up 15 times in a row?

(e) After playing this game times, how much do you expect to have?

(f) When would you stop playing this game? Put yourself in this situation and answer according to your personality. Then, find the "correct" answer using mathematics.

(g) Instead of sequentially rolling one die times, imagine rolling dice simultaneously, once. Would any of your answers change? Why or why not?