is the EV for this game really infinity?

[pococurante]

Actually I just used the graph at http://en.wikipedia.org/wiki/St._Petersburg_paradox for an estimate for 20k hands. I also found another graph on a second site with similar results, so I assumed it's fairly accurate. (And now I see that the game being used on there gives $1 for an immediate loss, while the game the OP used gives $2 for an immediate loss, so all my numbers are completely wrong. sorry.)

As for losing at first, but then going up later, that's because of the infrequency of the large wins.

If you pay $6 to play the version that starts at $1, chances are you will be losing for quite a while. Your first handful of games will probably be a bunch of 1s and 2s with an occasional 4 or 8. Only after a few thousand games, when you will have come across some large "jackpot" wins, will you become profitable in the long run.

[pococurante]

OK, here's how to get a realistic EV for a certain number of games. This method assumes average luck.

For 2,048 games with a $2 minimum prize, you will typically win the sum of:

1,024 wins for $2
512 * 4
256 * 8
128 * 16
64 * 32
32 * 64
16 * 128
8 * 256
4 * 512
2 * 1024
1 * 2048
(there is one game remaining, which is your "luckiest" game of the session, which is technically worth infinity. For the sake of convenience I'm calling it a win of 4096... yes I know this isn't mathematically correct)

Note that all of the above equations equal 2048. 2048 = 2^11. Adding in that final game, your total winnings will be 2048 * 13... meaning that your average per game will be $13... so all you have to do is find 2^X for your particular number of plays, then add 2.

# of plays ------ EV per game
8 --------------- 5
256 ------------- 10
1,024 ----------- 12
8,192 ----------- 15
32,768 ---------- 17
262,144 --------- 20
8,388,608 ------- 25
268,435,456 ----- 30

These numbers match up well with the graphs I've seen, so I'm pretty sure these are reasonably accurate. The EV rises quickly early on, but then slows down considerably. It takes only 8 games to reach an EV of 5, but then each increase of $5 EV requires 32 times as many games as you've played to that point.

Here's a quick chart... I don't have Excel installed so I used the retarded Google version that doesn't let you choose which info goes on which axis.


BTW I don't know why peterchi's results for $20 games didn't become profitable after 5 million games, it makes no sense for that to happen. It should fall sharply, slow down and hit a low point at ~110k games, then begin an upswing and break even at 262k games.

Yes, I was this bored... once I got started I just wanted to "beat the game".

[adMIT defeat]

The amount of completely incorrect math/probability theory in this thread is astonishing.

[pococurante]

[ QUOTE ]
The amount of completely incorrect math/probability theory in this thread is astonishing.

[/ QUOTE ]

Other than assuming I could substitute a small number for something that is technically equal to infinity, was there anything in my last post?

[pzhon]

I typed out a detailed description of how to estimate the probability that you win less than $100 million in 5 million trials, but apparently I took too long so the forum would not accept my post. Usually, I save the posts to the clipboard before trying to submit, but this time it didn't work. This time limit is annoying. What benefits are there that are supposed to outweigh the many lost posts?

Anyway, the answer was 1.43%, within about a factor of 1.02. The probability that you win more than $100 million is about 98.57%.

[uDevil]

[ QUOTE ]
I typed out a detailed description of how to estimate the probability that you win less than $100 million in 5 million trials, but apparently I took too long so the forum would not accept my post. Usually, I save the posts to the clipboard before trying to submit, but this time it didn't work. This time limit is annoying.


[/ QUOTE ]

Agreed. For longer posts, I use notepad then copy the text into the form. Unfortunately, this screws up the formatting, so sometimes I just copy the text from the form to the Windows' clipboard before attempting to submit it. If the submission fails for some reason, I can paste it into a new form.

[cabiness42]

Don't try calculating the EV right off the bat. Calculate the number of expected coin flips the game will last:

1/2 of the time it lasts 1 flip
1/4 of the time it lasts 2 flips
1/8 of the time it lasts 3 flips
etc.

So, the expected number of flips is (1*1/2)+(2*1/4)+(3*1/8)+ . . .

That number equals 2.

So the excpected length of the game is 2 flips, and your EV is then 4.

[DrVanNostrin]

[ QUOTE ]
I typed out a detailed description of how to estimate the probability that you win less than $100 million in 5 million trials, but apparently I took too long so the forum would not accept my post. Usually, I save the posts to the clipboard before trying to submit, but this time it didn't work. This time limit is annoying. What benefits are there that are supposed to outweigh the many lost posts?

Anyway, the answer was 1.43%, within about a factor of 1.02. The probability that you win more than $100 million is about 98.57%.

[/ QUOTE ]
Next time, just hit the back button on your browser, copy your post, re-reply and paste.

[pzhon]

[ QUOTE ]
[ QUOTE ]
I typed out a detailed description of how to estimate the probability that you win less than $100 million in 5 million trials, but apparently I took too long so the forum would not accept my post. Usually, I save the posts to the clipboard before trying to submit, but this time it didn't work. This time limit is annoying. What benefits are there that are supposed to outweigh the many lost posts?

Anyway, the answer was 1.43%, within about a factor of 1.02. The probability that you win more than $100 million is about 98.57%.

[/ QUOTE ]
Next time, just hit the back button on your browser, copy your post, re-reply and paste.

[/ QUOTE ]
That was the first thing I tried, of course, and it didn't work. Browsers do not always save the information in textboxes.

[pzhon]

[ QUOTE ]

So, the expected number of flips is [2]

So the excpected length of the game is 2 flips, and your EV is then 4.

[/ QUOTE ]
Non sequitur, and your conclusion is wrong. The correct answer has already been given along with justifications and links to other discussions.