Roll a dice one million times...

Ok, I got this question during an interview:

Would you rather:

#1 Roll a die once and be paid a million dollars times that number?

or

#2 Roll a die one million times and be paid one dollar times the number of each roll?

Basically, which scenario will average greater return?

I'd rather roll once and be paid one million times the number. I'd win a minimum of $1m (very handy amount) and avoid the years of RSI related medical expenses that would result from rolling a dice 1 million times.

If you roll a million times, the average of all those rolls will be very close to 7. So figure if you take that route, you're gonna end up with $7mil, give or take a couple bucks.

But roll once and there's a chance of rolling a 2, and a chance of rolling a 12, as well as all the numbers in between.

You've got an equal chance of ending up with more than $7mil as less, and a 1/6 chance of ending up with a 7. I'd take my chances and hope for a 12. or even an 8! Of course, I'll be bummed out, but take the $2mil consolation prize.


Bottom line: You're talking to gamblers.

I think most of us would roll just once.

Wait a sec...

Quote: tekpunk

Roll a dice

FYI: Dice is plural. Die is singular.

Did you mean a pair of dice? (That's what I thought when I wrote my reply), or a single die?

If it's the latter, then rolling a million times will give you about $3.5mil. Rolling one die once, and you won't ever end up with $3.5mil. But the other stuff is the same, although halved.

I'd still say roll once, and think most other gamblers will also say to roll just once.

Greater return? Neither. One affords you the opportunity to gamble with equal chances of all six amounts, while the other will give you a win that is very close to 3.5 million. Both offers you the same average return of 3.5 million.

Assuming I didn't have to physically roll the dice a million times and it was done with a computer, I'd probably take the guaranteed 3.5 million. Otherwise, I'd obviously roll it once!

I'll take the guaranteed amount. I need more than $1 million to get me to the end of life. If I was in a place where I only needed $1 million dollar to be very comfortable, then I would roll the dice and take a chance at $6 million.

The expected value of both is the same.

Using the decreasing value of money towards happines theory (whose name I just invented, but paraphrases the Wizard's suggestion that the value you get from additional money is proportional to the log of the amount), rolling 1 million dice is clearly superior

(also assuming you don't have to physically roll the dice).

Now that I fully understand the question.....


In my original reply, I thought you meant two dice. As such, the odds are good that the roll would be a 7, making it the same as rolling a million times. The odds are high of rolling a 6, 7, or 8, which, for me, is close enought to 7. So I'd roll once and hope to get at least a 6.

But now that I've thought about it, rolling only one die gives you an even chance of getting any amount.

There's to much (is variance the correct word? how about volatility?) for me.


I'd roll a million times.

Quote: dwheatley

Using the decreasing value of money towards happines theory (whose name I just invented, but paraphrases the Wizard's suggestion that the value you get from additional money is proportional to the log of the amount), rolling 1 million dice is clearly superior

(also assuming you don't have to physically roll the dice).

This is called the "utility of money" in economics. It is a concave curve (in the positive region of the Cartesian plane). Just like you say, each additional dollar has a lower utility than the previous.

For me, even though I enjoy a fair gamble, I would pick to roll 1 million times because of the above. Think about it, a guaranteed 3.5 million would mean you would never have to work again in your life, ever. You could live off the interest, even at some terrible rate, like 4%. You could not do this with 1 million.

Let's say you could roll a die and receive $1,000,000 times the amount rolled. Obviously the expected win is $3,500,000. What amount of money would you be indifferent to if as an alternative settle price? As someone mentioned, I claim the utility of money is roughly equal to the log of the amount. Assuming you had zero money to begin with, that would make the indifferent settle price $2,993,795. However, if I were really in that situation, I think my indifference point would be a bit less, like about 2.7M, not facting in the progressive US income taxes.