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Prime

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1. Half as old as my brother

To put a finer point, if two siblings are two cells (or particles) resulting from splitting of the mother cell into two, then none is older.

3. A lion and its tamer

I like happy endings. And that uncomplicated trajectory is nice too.
4. A lion and its tamer

I thought lions could pounce. But even if that is not allowed, the lion still must eat.
5. perfect powers

Barring complex numbers and limiting ourselves to integers...
6. A lion and its tamer

Why should a hungry lion devise the best strategy of escape for a tasty tamer?
7. A lion and its tamer

I be the Lion. There. Someone solve it for me. BTW, it was an Ogre last time, not a dog.
8. Comparing gambling systems

There is that little technicality in the above post. And then there are couple other points to make. Suppose, each of the 6 individual variants deviate within a chosen interval with a probability approaching 1. Don't we need to show that the ratio of all combinations of the deviating variants to all possible combinations (of 6N, where N tends to infinity) also approaches 1? And then, of course, that “Law of Big Numbers”. Who passed, that law? When was it ratified? How is it enforced? The entire proof is riding on it. But, I suppose, that is beyond the scope of this forum, unless there is some clever and funny way of demonstrating that Normal Distribution thingy without using any integrals. So never mind that. If I accept the proof for ending up with nothing after riding my winnings infinite number of times, there are only few differences remaining in the interpretation of the OP and gambling philosophies. 1) Does BMAD's casino compel its patrons to play forever once they started? I hope -- not. At least the OP does not mention it. (Could be in the fine print, though:) 2) While I am being held to playing a pure form of riding the winnings (Post#38), somehow, Rainman is allowed a modification of betting just a half of his bankroll on each turn (Post#40). Which is a very good prudent winning form, but I like my modification (Post#33) with the minimum bet requirement of \$1 and a bankroll of \$20 better -- it seems faster and more exciting. 3) Suppose we have limited time, limited resources, and BMAD allows his patrons to leave the game whenever they want to cash in their winnings. Then what is better: showing up at the casino with a bunch of singles and betting exactly \$1 on each turn until closing time; or walking in with just \$1 and riding your entire bankroll on each turn until satisfied with the amount won (more than a million) or rolling the die for 1000 times, whichever comes first? The riding seems more appealing to me. And 1000 consecutive rides look very good. After all, it is a long long way from 1000 to infinity. With 1000-roll riding, you cannot lose more than \$1; your overall chance of ending up ahead looks like 48.78%; and your chance of winning more than a Million \$\$ looks like 4%, or better*. Whereas betting \$1 at a time, you are basically working for \$33 all day long. So choose. *My simulation does not stop at the Million \$\$, but keeps rolling until 1000 rolls are done. In so doing it loses in some cases the millions procured in the interim, or wins some crazy amounts to the tune of 1012, which BMAD's casino is not able to honor. Again, I invite anyone, who feels up to the challenge, to post theoretical justification (or disproof) to the statistics of a 1000-roll ride presented here.
9. Comparing gambling systems

I like the reasoning, but the proof by example uses a slanted interval. The true Median of 1/6 is not in the middle of the interval 0.1666 to 0.16667. What happens if you use a truly median interval, e.g., from 9/60 to 11/60? It seems to be leading to the opposite conclusion.
10. Comparing gambling systems

What makes these topics so interesting to me, is my lack of education. Math education in particular. Perusing your argument from the post 24, imagine a casino offered you a choice between two games, 1000 dice roll each: 1) Stake \$1 with a 5% chance of winning \$1,000,000 or more; OR 2) Deposit \$300 with a 50% chance of winning something between \$16 and \$50, while in the extremely unlikely cases losing your entire \$300 or winning \$500. Which game would you play? (The percentages here are illustration -- not an actual calculation.) I think, what we are arguing here is called Normal Distribution, or Gaussian Distribution, or Bell Curve, or some other such name. In view of the above, I think, the wining likelihood of a 1000-roll ride is being underestimated.
11. Comparing gambling systems

But I have run a bunch of simulations. And some numeric analysis as well. Enough to convince myself. See my posts 12, 20, 22, and 33 inside the spoilers. Riding your winnings for 1000 rolls, seems like a good choice. That won millions in my simulations in very short order of time. At this point I feel that for further discussion, we need to solve analytically what is probability of winning X \$\$ or more while riding for N consecutive rolls. E.g., \$1,000,000 or more after 1000 rolls. For that one must find exact number of 1000-roll variations yielding more than 1,000,000 and divide that number by 61000. I don't feel up to the task at the moment. But I can provide any such data for up to 12 rolls.
12. Comparing gambling systems

When playing indefinitely, you cannot cash in your winnings and spend the money on this side of Infinity. And on that other side, who will care how much you've won, or what fraction of \$1 you have remaining? Strange things happen at the infinity: 1. You are expected to end up with nothing, if you ride your winnings infinite number of times. 2. That is because the most likely outcome is the one where each of the six possible roll values will have appeared the same number of times. The probability of that most likely outcome is zero. (That requires a proof though. See the spoiler.) 3. Your average payoff after infinite number of rolls should be infinite, if you bet just \$1 on every roll. 4. Your average payoff if you ride your winnings is going to be even more infinite. In fact, it is going to be infinitely greater than that from the previous point (3). That is because when riding your winnings, the infinite variable is an exponent whereas when betting a fixed amount on each roll the infinite variable is a multiplier. 5. To see your average payoff, you must play infinite number of sets of infinite number of rolls each. You need not live forever to go on that gambling binge. Just roll an infinite number of dice every time and roll them infinitely fast. Coming back from the Infinity, as I said here before, for all practical purposes, it is impossible to lose in this game. And you must ride your winnings, if you want to win big. The Expected Value (average payoff) for a 1000 consecutive rolls is (1.0333...)1000 = 1.74*1014. You cannot run enough experiments to confirm it empirically. Nonetheless, the winnings are good. In order to help its patrons to win large sums of money, the casino added a small modification to the game. Namely, the minimum bet requirement of \$1. So while rolling the dice and riding your winnings, if your total falls below \$1, you must add enough cents to make your next bet at least \$1. To sum up: In this game having a bankroll of \$20, and in reasonable amount of gambling time (few days) 1) If you ride your winnings, you most likely end up with the winnings of millions of \$\$. 2). When betting \$1 at a time, in the same amount of time, you'll most certainly walk away with few hundred of \$\$. (Frankly, not fun and not worth your time.)
13. Comparing gambling systems

+1 Precisely. For \$1 you'd be buying \$5 worth of chances (provided the pot can't be split between several winners.) That's how I play lottery. In gambling EV is what counts. I think, you've misunderstood what I was asking to prove. With N rolls there are 6N total variations. There are TW variations ending in a payoff P > 1 and TL variations ending with the final payoff P < 1. TW+TL=6N. Find the limit of TW/6N as N tends to infinity. I agree that "Expected Outcome" tends to zero as the number of rolls N tends to infinity. I just don't see it all that relevant to winning in this game. If this game was offered in casinos, I would have most certainly won millions in a matter of days starting with a bankroll of \$100 or so. And I wouldn't let the entire \$100 ride in just one sitting. Money management still counts. However, exponential accumulation of winnings should be the predominant scheme to win big. Another sure thing is -- the Casino offering that game would go bust very quickly.
14. Comparing gambling systems

Upon further reflection, I am leaning back to my first post(#3) and the solution therein. The Expected Outcome and Geometric Mean have no bearing on the winning scheme you must adopt in this game. In particular, the Geometric Mean formula does not offer any tangible means of predicting an outcome of a gambling binge, whereas the Expected Value formula works just fine in practice. Once again, those who don't believe it's a winning game, can play the Casino side.