Ilmari Karonen
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 Apr 14 awarded Yearling Apr 11 awarded Nice Answer Apr 10 answered I don't understand what a “free group” is! Apr 9 comment Understanding Euclid's proof that the number of primes is infinite. Interestingly, it seems that (the translation of) Euclid's original proof quoted by @JeppeStigNielsen does contain an "unnecessary" proof by contradiction, precisely for the fact that (in modern terms) no prime divisor of $1+\prod S$ can be a member of $S$ (since if it were, it would divide both $\prod S$ and $1+\prod S$, and thus also their difference $1$, "which is absurd"). Mar 31 revised What is the difference between the three types of logarithms? fix math spacing (\mathrm -> \operatorname; just use \ln for simplicity) Mar 28 comment Roll two dice. What is the probability that one die shows exactly two more than the other die? The lazy answer is, of course, anydice.com/program/7ffc Mar 28 revised Roll two dice. What is the probability that one die shows exactly two more than the other die? fill in missing table cells (the Common HTML math rendered doesn't seem to do that automatically) Mar 27 revised Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$ add math formatting, fix typo in equation (1 -> i) Mar 24 comment Why are turns not used as the default angle measure? Or if you want to link to Wikipedia, en.wikipedia.org/wiki/Tau_(mathematical_constant) would probably at least be a better target. Mar 15 comment What is the probability that a man likes pink? (In fact, confusing the probability of an observation, given that an assumption holds, with the probability of the assumption holding given the observation is a well known statistical fallacy: en.wikipedia.org/wiki/Prosecutor's_fallacy) Mar 15 comment What is the probability that a man likes pink? What your formula is computing is the probability that 2 out of 3 men answer "I like pink" when they play the game, given that none of them actually do. This is not generally the same as the probability of none of the men liking pink, assuming that two of them say they do when the play the game. Of course, the two probabilities can be related via Bayes' rule, but this relationship will depend on the unspecified prior probability of $k$ of the men liking pink, before we know the game results. Mar 15 comment What is the probability that a man likes pink? [...] Also, for such "inverse" problems, where we wish to estimate an underlying parameter from noisy observed data, the estimate will partly depend on our prior assumption about how likely each underlying fraction would be. (We can try to avoid biasing the estimate by choosing a "neutral" prior assumption, but that's still an assumption.) As the group size increases, this sensitivity to prior assumptions will decrease. Ps. See also: en.wikipedia.org/wiki/Randomized_response Mar 15 comment What is the probability that a man likes pink? This is the correct answer in the limit where the size of the group playing the game tends to infinity (or, equivalently, in the theoretical scenario where we assume 2/3 to be the a priori probability of a randomly chosen player to say "I like pink" before they roll the die). If we take 2/3 to be the observed outcome of an actual game played by a finite group of men, however, the outcome will have some variance that will affect the estimate of the underlying fraction of men that like pink. [...] Mar 13 comment Is it possible to prove uniqueness without using proof by contradiction? @Voyska: To prove that there exists a unique object satisfying some property, you need to prove that if $x$ and $y$ both satisfy the property, then $x = y$. That is not (necessarily) a proof by contradiction. Mar 13 comment Simple method to solve a geometry question for junior high school student @Mick: Sure, that's essentially the answer given by MXYMXY and Ameet Sharma. But I don't see where you're using that in your answer. Mar 13 comment Simple method to solve a geometry question for junior high school student It seems to me that either I'm missing something crucial, or your answer has a gap. Specifically, while it's certainly true that $[m] = FX$ is constant, and that $t = FO \le s + [m]$, with equality only when $O$, $X$ and $F$ are collinear, I don't see anything in your answer to justify the assertion that having those points collinear would necessarily maximize $t$, given that $s = XO$ is not a constant. Mar 12 comment Probability that a person is infected if test is positive? @Maasumi: Thanks for catching that! Fixed. Mar 12 revised Probability that a person is infected if test is positive? add missing "not" per comments Mar 3 comment Does $0 < x < 0$ imply $x =0$? @AlexProvost: No, you're not. Although I suspect that that's merely a somewhat garbled recounting of something else the professor said, which may have made more sense. Feb 29 comment What is the limit of $\frac{cos(1/x)}{x}$ when $x$ tends to Zer0? Numerically plotting the function is always a good place to start. It should at least give a hint as to whether there is a limit, and if so, what it should be.