131–140 of 277 posts
Earlier quoted context omitted.
Why would you test it? Probability of two heads: p*p Probability of two tails: (1-p)*(1-p) Probability of head followed by tails: p*(1-p) Probability of tails followed by heads: (1-p)*p It's not difficult to notice that if you remove the first two, the last two form a 50/50 distribution
Some of us suck at math.
Earlier quoted context omitted.
Some of us suck at math.
Math is often much more fun and compelling for some people when you both theoretically prove something works and then also convince yourself of the same thing via a numerical experiment. I’m pretty good at math proofs (pure math PhD, wrote some books and papers), but I still love to do numerical experiments. It’s fun, and you also set yourself up to be able to easily ask different questions that may be very hard to a…
Earlier quoted context omitted.
I'm confused, how does this help? If coins are biased to land same-side up, then don't I always have an advantage by guessing whatever side is up before the first throw?
I see what you're getting at, and it's subtle! To simplify the discussion, let's assume we always start out with heads up. You're right that the first coin is more likely to end up heads. But so is the second coin, and if both occur, that would invalidate the pair of tosses. Now, imagine you guessed tails despite the coin starting on heads. If the first toss lands tails, the second coin is still more likely to land h…
How is that possible when both outcomes have a probability of exactly 50%?
An ideal coin flipped by an unbiased super person would have that 50% probability. A real coin flipped by a real human no, as the study shows. Let's consider a spherical cow...
Earlier quoted context omitted.
An ideal coin flipped by an unbiased super person would have that 50% probability. A real coin flipped by a real human no, as the study shows. Let's consider a spherical cow...
It's simpler: an ideal coin flip is simply assumed to be uniformly distributed, on the basis of there being two possible outcomes and no influence. Where the bias in reality comes from, doesn't matter. This also happens to be the great divide between frequentists and Bayesians.
What's necessary to guess 50 % on the first toss is simply (a) complete ignorance about the bias, whatever it is, and (b) the hypothesis that the bias is just as likely to be negative as positive (i.e. a symmetric prior.)
Von Neumann described a very elegant way to get fair results from a biased coin. 1. Flip the coin twice 2. If you get the same result both times, goto 1 3. Now that you have different results for your pair of flips, use the first element of the pair of flips as your result. https://en.wikipedia.org/wiki/Fair_coin#Fair_results_from_a_...
I absolutely love that even with a corrupted system you can use properties of the system to ensure just outcomes.
I think you can argue that the experiment wasn't representative of 'normal' coin flips. On average, each flipper in that experiment flipped a coin over 7000 times, after that amount many people will have learned to flip in a comfortable way with less variance between physical action and force they use. I'd imagine that in that case the coin would more likely land with the same orientation. I don't think this would be…
(I know that there are techniques for adding the wobble to the toss, but I didn't study them and I have no clue how to do them. I think it is safe to say you don't discover them intuitevelly.)