Prometheus

I'm not entirely clear exactly which is the exposure and which the outcome, although you could guess from the wording of the question. Are these terms familiar to you? Is the 'correct' answer 0.70 by any chance?

I largely agree too, but where we disagree is interesting. Secular morality exceeded most religious morality some time back, and is now a hindrance to human progression. That is not the same as religion being irrelevant. It is relevant for two reasons. One, which we have touched upon, is that they contain the history of much of modern morality. By analogy, the events and people that have shaped your personal morality are not just some things from your past that you can do without: they fundamentally shaped how you make moral decisions today. So too human history has shaped our morality today, and understanding this history helps us make sense of how we came to our current state and how best to proceed. But perhaps more importantly, the vast majority of the world is religious and uses religion to make ethical decisions. It say religion is irrelevant to any moral discussion is to say the majority voice on Earth is irrelevant because a minority has exceeded such retrograde thought and knows better. In my opinion such sentiments are partially responsible for the wave of populism sweeping through the Western world: many people are getting sick of being considered irrelevant. Instead of telling people they are irrelevant we need to find some way of engaging with them, as unpalatable as that may seem. This means learning to engage with religion as it seems many of the disenfranchised seem to be religious.

Try spending hundreds on some quack medicine and you'll soon be trying to convince yourself it was worth it. No really, i did feel some benefit. Really i did. Honest, look i couldn't bend my arm like that before. Well it works for me and that's all that counts...

I agree, but the 'nothing but' is a bit misleading. We all like to believe modern morality is so obvious it couldn't ever have been otherwise. But our morality has taken millennia to develop: there is nothing obvious about it. Discussions thousands of years ago about rape say, were likely as loaded as discussions today about say, euthanasia (consider marital rape is still legal in many places, and only recently illegal in Western law). It is similar to science in that it has developed slowly based on the thoughts of proceeding people and a little more quickly when the occasional person who had deep insight pops up (to be clear only in that aspect do i believe it is similar to science). The Old Testament states an eye for an eye, but it was an improvement to older ethical systems in which the lose of an eye could be repaid with death or more: it states no more than an eye for an eye. Whether we like it or not, and whether for good or bad, religion has contributed significantly to the modern moral landscape - which is one reason why i believe religion is still relevant to modern morality: if only as a reminder of where we have from. I thought the US government was areligious when it first began?

I see now. Not necessarily wrong then, just different. Although if we then sum my sequence up to each n shouldn't we recover your sequence? Also, we know from that link that the sequence should sum to one - we could use this knowledge as one test of any closed form solution we make. Yes i did it manually which is why i only went up to 7. For this method to really work you really need to get a computer to perform the experiments - no errors (assuming the code is OK) and much quicker. This is essentially what Monte-Carlo simulation is: it's is used all over the place when analytic solutions to problems aren't forthcoming for whatever reason. Even if analytic solutions are available its often quicker to simulate the results, but arguably you lose understanding that way.

The researchers are from Harvard’s Osher Research Center, also known as the Osher Centre for Integrative Medicine. Integrative medicine is also known as alternative or complementary medicine. Some alternative practitioners are happy to admit that alternative therapies work purely by placebo: they may have an incentive in showing that placebo works whether the patient realises this or not. You do a grave disservice to both science and buddhism. But that's another topic...

It's a contentious claim: there is only that one study backing up the claim, and their results are less dramatic than most accompanying headlines (as usual). This blog explores the claim.

You may be unaware that google personalises searches so there is no guarantee we will recover the same page as you.

Well lets try your method and see. I'm starting from step one and trying to find the probability of being on step 0 after n moves. Also assuming 50/50 chance of moving backwards or forwards. Trying it for up to seven steps i get the sequence of probabilities for n=1,2,...,7 to be 1/2, 0, 1/8 ,0, 1/16, 0, 1/32. So it seems like a pattern is emerging, but i'm doing this 'by hand' so we must always be wary of mistakes. This is different to your sequence which doesn't seem right to me. Assuming this pattern continues (should really do some more steps but even by seven it was getting tedious - i don't think it will actually hold), we might say the probability that we get to step 0 on move 9 is 1/64. We want to find a formula that returns 0 on even n and returns certain powers of 2 for odd n. We then want to test it. A slightly better way might be to notice where this pattern is coming from: for any move n, the number of possible paths (to anywhere, not just 0) is [math] 2^n[/math]. Then we want to 'keep' just those paths that lead back to zero. So for n=7, we have [math] 2^7 = 128 [/math] and i counted 4 paths that lead to zero out of these 128 giving us [math]\frac{4}{128} = \frac{1}{32}[/math]. I can't think of a simple way to count all the paths to zero in n moves (without landing on n in any proceeding moves either) though. Does any of this make sense/help? Essentially you are adopting an empirical approach to calculating probabilities which is prone to mistakes, cannot be easily generalised (if we want to make the flips p/q instead of 50/50 for instance), and ultimately cannot actually be proved for all n without using the other methods anyway.

I'm sure we can all agree that there are good and bad people who are christian or muslims or whatever. The pertinent question is whether an ideology makes for a more intolerant/violent society. Again, i'm sure we all agree ISIS is an abomination to humanity, but does Islam encourage this type of ideology more than other religions? I think not compared to other monotheistic faiths, but it is difficult to tell. However, someone should not be labelled a bigot just for asking the question and seeking data, which does occur under the guise of 'intolerant of intolerance', as has happened to Sam Harris for instance.

If an effective treatment is available then it may be deemed unethical to withhold that treatment to give either untested or placebo medication. Some trials will test the untested med against this known treatment instead of placebo. Perhaps the phenomena the OP is referring to is conflating this practice? Which is why the clinicians are blind to whether treatment or placebo is being administered in the vast majority of trials. A trial needs a very good reason not to double-blind, otherwise it will not even get past set-up.

He won that 'odd' game. It was against the weakest opponent in a room full of top players.

Yeah, things can get strange with infinity. That's why it's hard to get an intuitive feel for it. Nothing wrong doing it the old fashioned way. There are stories of gamblers who would have a good idea of certain probabilities just by playing so much and then seeking confirmation from mathematicians. I was hinting at the phenomenon that you will never reach step zero, starting from step one, on an even number of attempts. Try it.

Yes. There's loads of subtle ways to change the set-up that can change probabilities drastically. Consider boundaries a and b where they are, say, 10 steps apart and we start in the middle of them. Whats the likelihood of staying bounded between them after one step? Zero obviously. So too for 2,3 and 4 steps. But after 5 steps there's a 1/16 chance of us hitting a or b. After six steps it's even more likely. And more and more likely for successive steps until, in the limit (i.e infinite steps), the probability is 1. Yes it's a bit different to Shakespeare's monkeys (though you can model both with a Markov chain). I was just trying to convey the point that any event with a finite probability of occurring. however small, will occur if given infinite chances. Yes, that's right. We can quantify things like the expected number of flips to get to this state. That number of flips is be very large, but since we have infinite flips, it will happen sooner or later. You are trying to calculate the probability of arriving at zero on step number n in that set-up aren't you? You tried this manually? Did you notice anything odd about how many steps it took to get to zero?

Yes. Yes, though there may be a slight difference in set up when you have a finite ruler and i have a finite bank balance: when you reach the end of your ruler you say i go back on a tail and stay still on a head, whereas when the bank has run out of money the gambler just walks away laughing and the game stops. It's not possible that the money will indefinitely fluctuate between any two values (except the lowest and highest values if those exist), but the probability of it fluctuating between two values for some number of flips can be calculated. There is nothing intuitive about infinity as far as i can tell. All i can say is that given that an outcome is possible (no matter how unlikely) and given infinite repetitions that outcome will occur. Like monkeys writing Shakespeare, notwithstanding practical limitations, they will reproduce all the sonnets. So is mine, don't worry about it. I find interest and effort to be more important to learning than any innate ability. You were making a start in the finite case, but here's something to think about: what is the probability of landing on step 0 given any even number of steps (and that you start on step 1)? P.S. What about programming? We can explore these concepts using Monte Carlo simulations.

This is a description of the Gambler's ruin and has been thoroughly studied, but is far from easy so no surprise you haven't got it first go. The Gambler's ruin is essentially a series of bets a gambler makes against a bank with infinite money: heads the gambler wins £1, tails the gambler loses £1. Even if the coin is unbiased , and no matter the gambler's starting stake (as long as it's finite) the gambler will loss all his money with probability one because the bank can always keep playing but the gambler cannot play after hitting zero as he can't afford to pay. Specifically what you are after is called the ruin probability. The proof is quite involved (P26) depending on how much maths we already know; we'd have to go through some preliminary learning first if you want to derive it together?

So i understand that two particles are said to be entangled if their joint wavefunction cannot be written as a product of their separate wavefunctions. So [math]\psi(\underline{r}_1, \underline{r}_2) = \psi_1(\underline{r}_1) \psi_2(\underline{r}_2) [/math] is not entangled whereas [math] \psi(\underline{r}_1, \underline{r}_2) = \frac{1}{\sqrt{2}}\Big(\psi_1(\underline{r}_1) \psi_2 (\underline{r}_2) - \psi_2(\underline{r}_2) \psi_1 (\underline{r}_1) \Big) [/math] is entangled. Does this stem from the fact that if two random variables in a joint probability distribution are independent of each other then their respective marginal distributions can also be written as a simple product: and if they cannot be expressed as products then they are not independent?

Cheers. I think i'm fine with the base vectors notation, not sure about the other stuff though. Going to ask some question on entanglement that might put it to the test.

All medications have side-effects. The decision is one of risks vs benefits: are the negative consequences of a medication preferable to whatever ails you. That decision is ultimately personal, the most extreme example probably being chemotherapy for cancer.

Indeed. I'll put this here in case anyone hasn't seen it yet.

As i said, if that is what you require to believe in order for love to make sense to you then so be it. Others have tried to give you evidence for this wonderful manifestation, but your heart and mind remain closed so there is no point discussing it. Never-mind, so long as your heart is open to love it doesn't really matter.

Evidently. Wonderful, isn't it?

If that is what you require to believe in love then so be it.

We should depend only on ourselves and each other, because love builds strong foundations.

Sounds good. No need for God.