SCENARIO: Gambling Gus offers daily unrelated longshots.

[edrugtrader]

Gambling Gus has decided to come up with a new 19:1 bet every day. These odds are exact, and all of the bets are fair. He offers you the long shot every day, and will pay you 19.01 units for a 1 unit bet. You have a large bankroll and could easily afford losing 1000 units.

Should you take this bet every day?

Does the fact that the true odds of each event being exactlly the same then group the events and make them related, such that the result set of all bets would normalize? Should I expect to make .01 units for every 20 individual bets I make over the long run?

Remember we are making each individual bet ONLY ONCE, and then betting on something completely unrelated with the same odds the next day.

This example was spurred by the discussion in this thread in the hold'em section, and I am curious about the statistics behind it.
HOLDEM THREAD (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=1120213&page=0&view=e xpanded&sb=5&o=14&vc=1)

[Richard Berg]

You're really asking two questions.

Mathematical independence: it's pretty simple. If the results of event #1 don't affect the probability distribution of event #2, then they are independent. End of story.

Whether or not you should take the bet depends on its standard deviation, your risk aversion, and your opportunity costs. Assuming they're independent, the EV is the same every day, but there may be more +EV / -SD things you could do with the same amount of time & bankroll. Then again, even if they're not independent, the covariance is just another factor you'd have to roll into your daily calculations, so it doesn't really matter.

[daryn]

take the bet!

[Richard Berg]

In the spirit of Ed Miller's smackdown (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=1122424&page=0&view=colla psed&sb=5&o=14&fpart=1), let me repeat that last sentence as clearly as possible: it doesn't matter whether your friend is offering unique 19:1 bets every day or simply rolling the same 20-sided die over & over. Being picky, if it does matter, it's not for the reason you think it does.

[soah]

http://www.google.com/search?num=100&hl=en&lr=&safe=off&q="kelly+criterion"&btnG=Search (http://www.google.com/search?num=100&hl=en&lr=&safe=off&q=)

edit: my browser screws up if I click the link, but copying/pasting works

[edtost]

browsers view the quotation marks in the url as the end of it.

edit: link removed, forum software won't stop changing % 22 to "

[TomCollins]

1000 units is not a safe bankroll. If I play until I double my bankroll, or go broke, I will go broke about 45% of the time. (I created a simulation for it). Even with 2000 units, and playing until I double, I was only doubling up 43% of the time before I went broke.

With a bankroll of 10,000 units, I am much more of a favorite to double my bankroll.

The results WILL normalize, but in the long run. The longer shot the probability, the LONGER it is. Suppose you had a lottery where you had a 1 in a million shot of winning. But the prize was 1 billion units. This is a great game. The problem is, you have a very good chance of playing it 1 million times without winning anything. So even if you have a bankroll of 1 million units, you have a very good chance of never winning it.

On your example, the EV sucks. It is only .0005. This makes the EV to variance giagantic. So even 1000 units is not enough to get you into the long run.

[Stork]

I think you can group them, as long as they have the same chances of hitting. That particular example, however, I'd pass on just because the EV is so slight that it's not worth the time.

[Bulldog]

[ QUOTE ]
Should you take this bet every day?


[/ QUOTE ]

Only if you have a safe full of cash (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Number=1121365&page=0&view=c ollapsed&sb=5&o=14&vc=1).