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# Staying alive with an eight-sided die

Best Answer CaptainEd, 30 January 2013 - 06:35 PM

Spoiler for Here's an argument

Go to the full post

16 replies to this topic

### #1 bonanova

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Posted 29 January 2013 - 04:43 PM

You glue two small square pyramids together at their bases to form an eight-sided object that becomes a fair die. You mark the opposite pairs of faces with 0 1 2 and 3.

You begin the game with \$1. You roll the die; the number that shows is your new stake. That is, with equal probability you lose you dollar, you keep your dollar, you double your dollar or you triple your dollar.

A minute later, you bet each of your dollars, if any remain after the first roll, with another roll of the die; one roll for each dollar that you have, and collect your winnings, if any. After another minute passes, each of your dollars, if any, suffers the fate of another roll. To be clear, in all cases each dollar is wagered individually. As the minutes turn into years, you eventually become rich, or you go bust.

What is the probability that you go bust?

I like the conditions that Prime gave in his last game. You can simulate the game if you like. But a submitted solution must comprise an answer and a method, both of which are correct. Enjoy!

• 0

Vidi vici veni.

### #2 Prime

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Posted 29 January 2013 - 11:09 PM

Am I allowed to stop betting and leave at any time?

If so, will they exchange my dollars for larger denominations, or write a check?

• 0

Past prime, actually.

### #3 bonanova

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Posted 30 January 2013 - 12:42 AM

Am I allowed to stop betting and leave at any time?

If so, will they exchange my dollars for larger denominations, or write a check?

Of course.

But your decision must be justified by a correct estimate of failure.

BTW, your winnings are all in Brain Den currency.

Send Rookie a PM with the address to send the check.

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Vidi vici veni.

### #4 Prime

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Posted 30 January 2013 - 01:22 AM

There is one thing that is not quite clear to me. The one minute delay is it between the rounds or between the individual rolls?

Say, after the first roll I got \$3. A minute later, do I get 3 rolls (one for each \$) in quick succession, or is there a minute delay before each individual roll?

• 0

Past prime, actually.

### #5 phil1882

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Posted 30 January 2013 - 04:14 AM

Spoiler for seems to me

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### #6 phil1882

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Posted 30 January 2013 - 04:27 AM

Spoiler for probability of going bust

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### #7 CaptainEd

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Posted 30 January 2013 - 06:18 AM

Phil, how does your progression incorporate the population increase?
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### #8 phil1882

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Posted 30 January 2013 - 06:30 AM

Spoiler for

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### #9 bonanova

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Posted 30 January 2013 - 08:35 AM

There is one thing that is not quite clear to me. The one minute delay is it between the rounds or between the individual rolls?

Say, after the first roll I got \$3. A minute later, do I get 3 rolls (one for each \$) in quick succession, or is there a minute delay before each individual roll?

The OP specifies that you get three rolls in that next minute.

Every dollar in your stake gets to be must be wagered each minute.

Each dollar thus has an experience that is independent of but statistically identical to all the others.

Does that suggest an approach to analyze the problem?

But the answer has to be the same as if you considered a simple infinite sequence of wagers.

Since all the events are independent, and we're looking for a stop condition.

So there are at least two analytical approaches to take.

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Vidi vici veni.

### #10 Prime

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Posted 30 January 2013 - 09:01 AM

In this case...

Spoiler for It's a living...

As for the question in the OP, I think I came up with a closed from equation for the losing probability, but haven't solved it.

• 0

Past prime, actually.

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