It costs $1 to roll a die, you win $6 for a 1. For max profit, ...?

Question

how many times do you play, and after winning how much should you stop?

Intuitively, I know these four answers are wrong...

Answer 1

This is a binomial distribution with a 5/6 chance of losing a dollar and a 1/6 chance of winning $5... winning six while paying the one to play. The mean expected value of wins after "n" rolls of the die is (n/6), and the standard deviation is [(5n)^(1/2)]/6.

The thing is, you're equally likely to lose money are you are to win money at this game.

When n gets really large, the expected value of your net winnings will move farther away from (n/6), but the variance and standard deviation will become smaller. For examples, it would take 8 rolls of the die to get a standard deviation of possibilities net wins of 1, but 29 rolls to get to a standard deviation of 2, 65 rolls to get to 3, 116 rolls to get to 4, and 180 rolls to get to 5.

In other words, the more you play, if you're on a winning streak and you play a long time, your winnings are more likely to increase, though at a much slower rate. But if you're on a losing streak, the longer you play makes it equally as likely to never catch up to break even.

The bottom line is the answer to maximize your profit depends on whether you're ahead or behind after your first few dozen throws or so. If you're losing, quit. If you're ahead, play 'til you're bored.

Answer 2

On average, I think you'd break even after every 6 rolls of the die. The best time to stop is when you've broken even after one roll and you roll a 1 on the very next roll. That way, you end up $5 ahead. I guess it would take about 36 rolls on average to do that, but I'm not exactly sure how to show it. (I assumed that 1 is the only face you get money for, by the way.)

Answer 3

For an even-weighted die, your probability is 1/6 for getting a 1.

From a finance background, your expected payout per roll is $6 * 1/6 or $1. Since each roll costs $1, your net expected result is $0.

So if you play, stop after your roll your first 1.

Answer 4

You play 6 times and you should stop at $10

Answer 5

so a million/2 the time you win $2 and a million/2 the time you lose $a million (assuming the die is truthful). so after 2 rolls you will anticipate to win $2 as quickly as and lose $a million as quickly as. this suggests ordinary you gained $a million. so the respond is $a million/2 rolls = $.50