For fun, I decided to calculate this exactly. The probability of busting on a bankroll of $10 within 60 trials is 39464863030417006100870577/3^60, which is .000930969… .
Ugly Haskell below (which solves this problem easily, but is not at all what one would use to efficiently solve it for much larger numbers of trials, should that need ever arise for you):
advance (busted:otherprobs)
= let ups = (busted:0:otherprobs)
downs = (busted + head otherprobs):((tail otherprobs)++[0, 0])
in zipWith (\a b -> 2 * a + 1 * b) ups downs
bustProb bankroll numTurns
= let initialDist = [if n < bankroll then 0 else 1 | n <- [0..bankroll]]
numer:_ = (iterate advance initialDist) !! numTurns
denom = 3^numTurns
in (numer, denom)
Of course: With a 2/3 chance of gaining $1 and a 1/3 chance of losing $1 on each flip, we have a probabilistic average single-flip profit of 2/3 * $1 - 1/3 * $1 = $1/3. And the number of flips you are taking is gargantuan relative to the variance per flip, so the overall profit over the number of flips is very unlikely to stray significantly from this.
But the computer is only doing in its simulations an imperfect approximation of something you might just as well do perfectly by hand! That’s always what’s weird to me about people doing these sorts of things, though I guess it’s at least more fun here than when people use it to convince themselves of Monty Hall or what have you. But different strokes for different folks.
And I suppose “imperfect approximation” presupposes a particular notion of what it is the goal of such simulation is; your goal might just be the entertainment of watching the dice roll, so to speak, especially in a situation like this where sampling one point from the distribution is potentially easier to grok than writing out the full thing. So, really, that last paragraph of mine should be rewritten or stricken… Please forget I said it.
I’m not certain if you realize this, or if it’s intentional, or even if it’s appropriate for me to so comment, but your posts occasionally seem to carry a condescending tone.
The part that I bolded is key. Analogy: the average person who walks into a Vegas casino is likely to leave with less money than he went in with. The longer he’s in there, the more likely it is that the casino’s odds will work against him and the more he’s going to lose. As his time in the casino tends toward infinity, his financial outcome tends toward the inevitable loss of everything.
Agreed. The central point here is that in the described situation the probability of the outcome at infinity is – far from being some meaningless unknown – actually very precisely defined. It’s like my previous analogy of integral calculus: it finds the area under a curve by summing the areas of an infinite number of infinitesimal rectangles, which turns out to be – not a meaningless number or “infinity” – but a precise measure of that exact area!
Yes, I did realize that, and that’s why I wished to apologize for that aspect of my post. It’s not intentional, but it’s certainly appropriate for you to mention it. I shall try to be better about avoiding that.
Hmm, on review I can see that I made quite the hash in my post 77 description of Gardner’s essay, no doubt from first trying to describe it from memory, than taking the trouble of finding the book for a more exact description, then sloppily merging the two.
For what it’s worth, I don’t see Indistinguishable as condescending: I see him as being obsessed with precise language, terminology, and expression, together with a bit of frustration at people not saying quite exactly what they mean. Which is a worthwhile trait in a mathematician, but I suppose can seem a bit harsh to a layman.
I have, frustratingly, argued that a casino would still win overall, even if every game were even and there was no house advantage in the per-play odds. Simply because, as a practical real-world matter, the casino has an “infinite” amount of money, and can play “forever” without going busted. Gardner appears to support that position.
Sure, but with a bias that is two to one in your favour, you’d be a fool NOT to play until you’d amassed a large enough pile that just sticking it in the bank at 1% gives you a faster return than playing for a buck at a time.
Probably not, but what they do have (and very wisely) is a limit to the amount which any individual player can wager on any particular event.
In Horace Levinson’s excellent tome “The Science of Chance” he recounts an anecdote wherein one of the Rothschild dynasty says to M. Blanc (the owner of the casino at Monte Carlo) “Take off your limits and I will play all day and all night”.
I disagree. Gardner’s claim depends on playing until one player goes bust. That’s not how casino games generally work in real life. The game ends when a player feels like it, or it’s time to go home – and this is determined by the player, not the casino.
Moreover, to paraphrase what Chronos said, while the casino effectively has an infinite amount of money, the world effectively provides an infinite number of players. If every game were even, over the long term the casino would probably just break even and would not be able to pay its bills.
Simply because in any single instance, the outcome might defy the odds.