Pattern in tossing a coin - which comes first? You toss a fair coin with sides $K$ and $Z$ infinitely often. What is the probability of the event that the pattern $KKK$ occurs before $ZKZ$ ?  I did the following: Let $(Z_n)_{n\in\Bbb N_0}$ be i.i.d. with $\Bbb P (Z_1 = K) = 1/2 = P (Z_1 = Z)$. We define the Markov chain $X_n$ by $$X_n = (Z_n , Z_{n+1} , Z_{n+2})$$ and $\alpha := \text{Uniform}(\{K,Z\}^3)$ and the entry times $T_A := \inf\{n\geq 0 : X_n \in A\}$ for $A\subset \{K,Z\}^3$. Writing in short forms, we are interested in $$\Bbb P _\alpha( T_{KKK} &lt; T_{ZKZ})$$ We can define the function $f(z) = P _z( T_{KKK} &lt; T_{ZKZ})$ and calculate by the first step method   $f(ZKZ) = 0$ $f(KKK) = 1$ $f(ZZZ) = f(ZZK)$ $f(ZZK) = \frac 1 2 f (ZKK)$ $f(ZKK) = \frac 1 2 + \frac 1 2 f(KKZ)$ $f(KKZ) = \frac 1 2 f(ZKK)$ $f(KZZ) = \frac 1 2 f(ZZZ) + \frac 1 2 f(ZZK)$ $f(KZK) = \frac 1 2 f(ZKK)$   Combining 5. and 6. yields $f(KKZ) = \frac 1 3$ and this makes the full calculation of $f$ possible. Thus  $$\Bbb P _\alpha( T_{KKK} &lt; T_{ZKZ}) = \frac 1 8 \sum_{z\in \{K,Z\}^3} f(z) = \frac 5 {12}$$  Is there a more elegant way to solve this?  Edit: I really ask for different approaches (not only one different from mine)
Probability of flipping coin What is the probability of obtaining THT before HHH when a toss is tossed sequentially? Could someone solve using state diagram and law of total probability?
How do I set a custom resolution? I live in Hong Kong, and the Play Store displays everything in the local Cantonese language. I have set everything I can (Android language, Account language, even the app locale via Xposed Module "App Settings") to English, but I still see all the categories in the store in Chinese. Can someone help me change those to English?  (Click image to enlarge) I don't understand, because if 'don't' is 'do not' then wouldn't the sentence without conjunctions be 'do not you dare' which I'm pretty sure isn't grammatically correct. Like, when we say the term it means something else, certainly not 'do not you dare' anyway. Any ideas on why? I am so glad English is my first language. How to find sum of power series $\sum_{n=0}^\infty\frac{1}{n!(n+2)}$ by differentiation and integration? Let $$ S = \sum_{n=0}^\infty \frac{1}{n!(n+2)} $$  Integrate the Taylor Series of $xe^x$ to show that S = 1  Also, Differentiate the Taylor series of $\frac{e^x - 1}{x}$ to show that S = 1  For the integration one, I got $\int xe^x dx$ = $x^2\sum_{n=0}^\infty \frac{1}{n!(n+2)} x^n$ and pretty much stuck. I can see the S term but i have no idea how to move on from there Why does my user number show up in the search box when clicking my profile information? The day before yesterday I think, my daughter asked me where we were going and since I was not planning to do the weekly shopping on that day and I just wanted to buy some stuff from two stores, I said offhand we are going to the shops! Looking back, it seems when I don't find a NAmE word I turn to UK if I know of.  So what do you say when:   go shopping (it's just go shopping!) you go to buy something specific; I mean you're not going shopping, and it's usually found in any store. You don't have to shop around. you want to buy non grocery or clothing stuff, maybe a can of paint and a spare part for your car together.    I have found some kind of answer for this question on , but I prefer to get a clear answer to my points above, and also if you agree with this native speaker about using store in singular form not plural:     Just as a side note, we don't use the plural in American English. "I'm   going to the store" (even if I may visit several stores), not "I'm   going to the stores". (If we know we are going to several stores I   think it would be more likely to say "I'm going shopping" or even "I'm   going downtown" in smaller towns.) The strange thing about the maximum in Planck's law I agree that every page in a site should have a unique TITLE, and that every page should have one and only one H1 element, but should the text of each H1 be completely unique too?  Instinctively one would think so, but there are potentially many cases where (IMHO) mind this doesn't make sense and attempting to do so will impair the user experience &amp; presentation of the page - for example, where pages are part of a closely related cluster, e.g. a multi-step process/tool/wizard.  Surely the process name should go in the H1 (e.g. "Pregnancy Planner"), and the current step you're on should be marked up as a H2 (e.g. "Step 1 - Trying for a baby"), rather than shoehorning both into a single H1 together (e.g. "Pregnancy Planner - Step 1 - Trying for a baby")?  Not only does this make the H1 rather long and potentially impact the presentation of the page, I would argue that in combining both into one you also lose semantic goodness, as the page is surely semantically richer by keeping parent (process) and child (step) as separate, nested headings?  Furthermore, the current proposal I have read is that on the first page the H1 should be the name of the process and the first step an H2, but on subsequent pages the same text should not be a H1 at all (just text styled exactly the same as an H1) and the current step should become the H1 instead. For me this is introducing unnecessary complexity to the page authoring/production process, and inconsistency in the semantic structure of the pages within the stepped process.  So it all comes back to the basic assertion that each H1 must be completely unique, which I question. Can you convince me that this should be so and that the cost/complexity of implementing this (especially in situations such as that above) is worth it?  Your thoughts please... Is there a way to see questions with a certain tag from all Stack Exchange sites? For example, I see there are questions with the  tag on Stack Overflow, but there are also such questions on Super User, Unix/Linux, and other sites in the network.  Is there a way to follow a tag across all of the Stack Exchange sites, so that one can see the global activity of a topic without visiting all the sites separately? I just had a case of a user who re-posted his question because his first question was closed as a duplicate. And, logically, I closed his re-post as a duplicate of his original, asking him to edit his original and explain why it's not a duplicate, so we can reopen it.  This process has worked like that for years now, but suddenly I realized that with the new message, question repetition is even encouraged:     This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.   Clearly, a new user who finds their questions closed will now be happy to just post their question again and again, until it's not a duplicate anymore, instead of just fixing their original post.  We need to be more explicit about this, and show this message to the OP of the question:     If those answers do not fully address your question, please edit your question and explain why it is not a duplicate.   The original message should still be visible to other users and visitors, as they can't fix that question, of course. How can I repair a ceiling fan that hums, but doesn't turn? Proof that all abelian simple groups are cyclic groups of prime order Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks!  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has either infinite order or even order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has odd order, and by Feit-Thompson, $G\cong Z_p$ for some prime $p$.  Edit: Thanks, I see that Feit-Thompson is too much.  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has infinite order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has finite order. Suppose $x$ has composite order $n$ so for some $p &gt; 1$ that divides $n$, $\langle x^p \rangle$ is a proper non-trivial subgroup of $G$, so $G$ is not simple. So $G$ is a cyclic group of prime order. I'm struggling to understand the core reason to invest in stocks as a minority shareholder. Majority shareholding makes more sense as you effectively control the company, so you can pay yourself dividends or even sell the entire company at will. You can guarantee your own payback.  However, as a minority shareholder, you don't get to make any of these decisions. You're basically just hoping to eventually, some day get paid, if the majority holders decide so.  Why do so many investors agree to pay now huge sums of money (trillions in total) for something that may eventually pay dividends, and even in the best case, these dividends are only a fraction of the price you paid to acquire these stocks?  The average annual dividend rate for the US stock market is 2%. That means if you bought 100 million worth of shares, you'd have to wait 50 years to recover your investment.  Of course, the most probable way you'd eventually get liquidity for these shares is by selling them to some other starry-eyed investor - which makes the whole thing seem like a giant Ponzi scheme.  Given all that, it's unclear to me why the public is willing to spend so much money to acquire minority shares in companies. What is the difference between  and != Intersection of powers of an ideal in a Noetherian ring Given a Noetherian ring $R$ and a proper ideal $I$ of it.  Is it true that $$\bigcap_{n\ge 1} I^n=0$$ as $n$ varies over all natural numbers? If not, is it true if $I$ is a maximal ideal? If not, is it true if $I$ is the maximal ideal of a local ring $R$? If not, is it true under additional assumptions on $R$ (like $R$ is regular)? Every season near the trade deadline, we see teams that are out of it trading expensive players and/or free agents-to-be, typically for younger players or prospects.  For example, Matt Garza, set to become a free agent after the season, was traded to the Rangers for a few prospects.  It made sense for the Cubs to trade Garza, because he wasn't under contract any further than this season, and thus doesn't have much value to them.  Compare Garza to someone like Mike Trout, who is likely the most valuable property in baseball, given his skill and inexpensive cost over the next several years.  There's no way the Angels would trade him, because he'll be the cost-controlled centerpiece of the team for years to come.  However, with the Angels out of it this year, what if they wanted to trade Trout to a contender (say the Pirates), but only for the remainder of this season?  That is, they'd get several top prospects now, and would have an agreement with the Pirates that they'd trade him back in the offseason (for something pretty worthless, say for a journeyman minor leaguer).  The Pirates get Trout for the stretch run and are happy to give up a few prospects.  Trout doesn't have much value to the Angels this year, but he has a ton of value for next year.  So they get a few prospects for him, in exchange for the part of his career which has very little value to them.  So in effect, the Pirates are renting or borrowing him for the remainder of 2013.  Why don't trades like this happen?  Are there any rules prohibiting trades like this? Where to put the periods when using a parenthetical sentence? I have two complete sentences; the second is in parentheses, essentially a comment on the first.  Where do I place the periods?  I would use this:     They played opera. (I covered my ears.)   Is this punctuation correct? I have calculated AIC and AICc to compare two general linear mixed models; The AICs are positive with model 1 having a lower AIC than model 2.  However, the values for AICc are both negative (model 1 is still &lt; model 2).  Is it valid to use and compare negative AICc values? Stacking Multiple Ternary Operators in PHP What to do with cross-site duplicates? What's the right course of action for cross-site duplicates?  There are two flavors:   A question is posted which belongs on another site and ends up already having been asked there. I can't find an example right now, but it's bound to happen soon if it hasn't already. The same question is posted in two places at the same time. Take  for example, which is a duplicate of , posted within a half hour of each other by the same user.   Possible actions:   Move it to the proper site and close it once it gets there as a duplicate. Provide a way to close it right away as a cross-site duplicate, without moving it first. Flag it and let a moderator merge them. Close it as "no longer relevant" or something else.   Additionally, flavor 2 is a bit of a special case; perhaps the question really spans both sites in terms of content and intended audience. Should multiple-postings in this case be allowed? If so, should they somehow be linked to each other? If not, how do we avoid a migration war?