Comments by "EebstertheGreat" (@EebstertheGreat) on "Wendover Productions" channel.

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  7. OK, I've come up with another way of explaining this. We both have conflicting claims, so we will perform an experiment to distinguish between them. Your claim: When low cards are more likely to get dealt in a game of blackjack, the player has better odds than when high cards are more likely to be dealt. My claim: the opposite. To test our claims, we need to actually set up situations where more low cards or more high cards are dealt measure the winnings in each situation. After many trials, if the low-card group wins more, then your claim is supported, and if the high-card group wins more, then my claim is supported. It turns out that this experiment has already been performed many, many times. If a shoe has more low cards than high cards remaining, then they are more likely to be dealt, just like if you were playing with a rigged shuffler. This happens precisely when the count is low. ALternatively, if the shoe has more high cards than low cards (i.e. when the count is high), then high cards are more likely to be dealt. So we compare the winnings in these two cases. And what do you know, players win more when the count is high than when the count is low. That's why they bet more on high counts. That's the whole point of the system. From this we can conclude that when low cards are more likely to be dealt in a game of blackjack, the player has worse odds as a result. Note that it doesn't matter why low cards show up more often, just that they do. This could be because there are more low cards remaining in a conventional shoe, or it could be because a continuous shuffling machine has been rigged. Either way, the result is the same. What about this explanation do you disagree with? (And incidentally, it is not true that cards just dealt are equally likely to come up next in order. For one thing, dealers do not usually put cards in after every deal but after a few. More importantly, most machines are actually not very effective at randomizing the deck, and in particular, cards just put in do not tend to go to the top. Some machines with small buffers have actually been exploited by card counters in the past with more success than against a conventional shoe. Some machines do not suffer from this problem, but none are perfect. But none of this is relevant to the rest of the thread.)
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  12.  @neumo5005  There is more than one way to define multiplication by infinity. On the extended real line for instance, which is the union of the set of real numbers with ∞ and -∞, 0×∞, ∞×0, 0×(-∞), and -∞×0 are all left undefined. Similarly, 0/0, ∞/∞, (-∞)/∞, ∞/(-∞), and (-∞)/(-∞) are all undefined, as are ∞-∞, ∞+(-∞), (-∞)+∞, (-∞)-(-∞), 1^∞, (-1)^∞ and 0^0. This reflects the fact that these are all indeterminate forms in calculus. That is to say, if we have functions f and g with lim f(x) = ∞ and lim g(x) = 0, the product f(x)g(x) may have no limit, or it may have any limit. We get similar results on the projective real line, which has only a single point at "unsigned" infinity. A very different notion of infinite numbers comes from Georg Cantor, where they are used either to compare the size of infinite sets or to label infinite lists. In these cases, any number (even an infinite one) multiplied by zero equals zero, by definition. The confusion comes from the fact that the idea of "zero times infinity" is underspecified. "Infinity" is too vague in this context. If we want to know how many elements are in an infinite product of empty sets, the answer is zero. It doesn't matter how many times you combine these empty sets, there will never be anything in any of them. But if we want to find the area of a shape by cutting it into infinitesimally thin slices, we are effectively calculating a sort of "zero times infinity" that clearly must have a positive result, since the shape has some positive area. And indeed it could have any area. Or you could try to calculate the area of the whole plane and get infinity. And there are even more pathological examples where you can't reach any answer at all. So it really depends on context.
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