What are the odds of rolling a perfect 244/251 Pass Line bets to achieve 1.41% house advantage in craps?

Right. Difficult to understand exactly where you're coming from.

When folks talk about the house having a 1.41% advantage on any particular bet, you can't ever expect that advantage to hold true for every set amount of bets.

In any 1,000 rolls you may find as little as 250 Pass Line wins, or you might find 700 Pass Line wins. You might even occasionally find more or less than those numbers. You might occasionally find a nicely spread out ratio, but probably not typically.

Just not sure if this comes close to answering your question. It's just too tough to understand where you're coming from.

***EDIT***

Quote: "So far there have been no answers that are correct or intellectually responsive." <---I'd say the same about your question, actually. Also, this is not exactly the way to inspire people to answer, but I'll add some thoughts in spite of that.

Quote: "The odds of having 1 loser and 494 winners are exactly the same or any other 495 combination." This is the reason I said that about your question.

The flaw in your question is thinking there are 495 combinations. To say there are more is the understatement of a lifetime. I'll share the number at the end, but for now let me show you a simpler example.

Let's just call everything a 50/50 chance for a minute just to simplify. Also to simplify, I'm going to ask you to imagine that there are only 50 series of "wins" and "losses" instead of 495. (I'll tell you the actual numbers when it comes to 495 at the end.)

If we play 50 times with a 50/50 chance, there are 1,125,899,906,842,624 different combinations of wins and losses!!! That's already 1.1259 QUADRILLION combinations!!! Now taking you logic, you'd say that there is just as much chance to win 1 and lose 49 as there are with any other comination. SORT OF! The problem is, out of the over 1 quadrillion combinations, there are only 50 of these combinations that involve 1 win and 49 losses! (There are another 50 combinations that involve 1 loss and 49 wins, too.)

Even with a series as short as 50, it is almost safe to say that nobody on this earth would ever see exactly 1 win or only 1 loss out of 50 in their lifetime in a 50/50 chance.

OK, so how about if there are 495 rolls in the series of rolls? Ready? Again I'm going to still simplify and call the odds of winning or losing 50%. The number of combinations of wins and losses after 495 rolls is 1 out of the number 1 with 149 zeros after it!!! Our wee brains can't even fathom this number.

Out of that enormous number of combinations, only 495 of them include just 1 win and 494 losses. Only 495 of those combinations include just 1 loss and 494 wins.

The only thing right about the assertions in your question is that this has never, nor will ever happen. Not in a million universes that are just like ours.

*****Finishing up - The House Advantage is 1.41%. This is a CONSTANT ADVANTAGE that the casino has over you at the start of EVERY roll. EVERY turn you're either going to win or lose. The wins and losses of any single turn, or any series of 495 turns have nothing to do specifically with the 1.41%. It's simply the "long run" advantage they will always have over you in the course of your lifetime of playing Craps.

It sounds like the asker is asking "What are the odds that someone will win exactly 244 out of 495 pass line bets?"

Well the probability of hitting a pass line bet is

49.292929%

The probability of losing a pass line bet is

50.70707%

(Note that the difference in the two probabilities is the house edge of 1.414141%)

So the probability of hitting exactly 244 wins in 495 bets is

= (0.49292929)^244 * (0.50707070)^251 * 495C244

= (0.49292929)^244 * (0.50707070)^251 * 495!/(244! * 251!)

≈ 0.0358475...

So there is a 3.58% chance that you will get exactly 244 wins in 251 pass line bets.

≈ 358 / 10000

≈ 1 / 28

So to answer the question, the odds of winning exactly 244 pass line bets in 495 attempts are approximately

27 to 1

It sounds like you (the asker) think that a house advantage can only exist if EVERY series of 495 come out rolls will result in EXACTLY 244 wins. You have to understand that the casino makes money over the long run and they don't need to win exactly 251 times for every 495 bets. Sometimes it will be more, sometimes it will be fewer.

When you say "I contend it has never been done or documented." you are incorrect. I contend that it is done once every 28 attempts or so.

When you say "The odds of having 1 loser and 494 winners are exactly the same or any other 495 combination." you are also incorrect. The probability of losing once and winning 493 times is

= (0.49292929)^494 * (0.50707070)^1 * 495C494

= (0.49292929)^494 * (0.50707070)^1 * 495

≈ 4.31796... x 10^-150

≈ 0.00000000000000000000

...00000000000000000000

...00000000000000000000

...00000000000000000000

...00000000000000000000

...00000000000000000000

...00000000000000000000

...000000000431796

which for all intents and purposes is zero.

I would, but I don't understand what you are talking about.

If the first 20 pass line bets have seven losers, then why does the next 224 bets have to be winners? The odds say 244 winners vs 251 losers, not 244 winners in 251 tries.

The average outcome in 495 attempts is 244 winners and 251 losers. That's just what it is, based on the dice and mathematics and probabilities. I'm not sure what you are arguing here. Are you suggesting the house advatange is higher or lower?

Then again, since you suggested that there needs to be 224 winners in a row, I'm not sure you have any idea what you mean.

EDIT: Sorry, you still aren't clear. What is your question? Are you suggesting that no single person is personally going to experience a 1.41% disadvantage because they are extremely unlikely to encounter 244 wins and 251 losses in a 495 roll span? So what? Just because one person doesn't experience that doesn't mean that's not the correct house advatange. That the average over millions of rolls.

EDIT AGAIN: I'm sorry that you aren't smart enough to realize your question makes no sense.

Boy...You have no idea how much I've thought about that question. We bought 22 acres about 7 years ago and built a tiny "mother in laws" house with plans to build a bigger home at a later time. We own two contracting companies, Natural Stone Fabrication and the other w/ a contractor's license does installations. We also own a nice Italian Restaurant. Need I say what the housing crisis has done to our contracting companies...ditto with the restaurant because NO ONE is spending right now. I would just build a one level three bedroom, three bath home with a huge backyard for the kids. Right now the kids share our bedroom and it's horrible. Our wood fence blew down during the Santa Ana's and we don't have the money to re-build it. I can't get a break from these kids to save my soul. I even had Mike cut my hair because it's too much of a hassle to get someone up hear to babysit for an hour to get my hair cut. YOU SHOULD SEE MY HAIR! OH MY LORD! *roflmao* I don't know if that's maniacal laughter or the laughter of lost causes!

Yo: Wins if the shooter rolls 11.

3 (ace-deuce): Wins if the shooter rolls a 3.

2 (craps aces): Wins if shooter rolls a 2.

12 (craps): Wins if shooter rolls a 12.

2 or 12 (hi-lo): Wins if shooter rolls a 2 or 12. The stickman places this bet on the line dividing the 2 and 12 bets.

Craps: Wins if the shooter rolls 2, 3 or 12.

0% you got it when you said "broken down" it is broken down from a computer simulation of a billion rolls of the dice... UNBELIEVABLE!!! while answering this question my sphinx cat just did it knocking around the dice...''purrrrrrrfectly''