Monty Hall Problem: Reasoning Behind the Logic I watched  from the MIT OpenCourseWare where the lecturer talks about the Monty Hall problem. He draws the decision tree and we find that actually you have a 2/3 chance of winning if you switch, and a 1/3 if you stick.  I understand the intuition, and the proof behind this (i.e. the decision tree).   However, what I dont understand is when a door is revealed, why doesn't the probability change from 1/3 to 1/2? Why is picking a door, then having one revealed, different from just having two doors and picking one at random?
The Monty Hall problem I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.  The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):  You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?  The character picks A, as the odds are all equally in his favor.  The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.  The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.  What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?  I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.  So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful.
random access file in java I have the following fields:   Inventory control (16 byte record)   Product ID code (int – 4 bytes) Quantity in stock (int – 4 bytes) Price (double – 8 bytes)    How do I create a fixed length random access file using the above lengths? I tried some examples online, but I either get an EOF exception or random address values when I try to access them.  I tried some more examples and couldn't understand the concept very well. I'm trying a project with it and will try to explore more on it.  Here is some example data. There might be holes in the data where No. in stock could be 23 == 023.            Quantity ID. No.   In Stock   Price  -------   --------   ------  1001       476      $28.35  1002       240      $32.56  1003       517      $51.27  1004       284      $23.75  1005       165      $32.25   Thanks for the help. Right identity and Right inverse implies a group Let $(G, *)$ be a semi-group. Suppose   $ \exists e \in G$ such that $\forall a \in G,\  ae = a$;  $\forall a \in G, \exists a^{-1} \in G$ such that $aa^{-1} = e$.    How can we prove that $(G,*)$ is a group? I'm moving all files to a new computer, via an external hard drive. I am certain that broken links will be a concern. What can I do to fix this? Is there a 'Best Practices' kind of way to prevent this problem beforehand? I have many hundreds of .blend files, and of course, I fear losing them. Some items in Terraria drop only very rarely from certain enemies. It's a pain having to farm so many enemies to try to get these rare items. Is there any way to make them drop more often? How can I remove a trailing newline? How can I draw annotated braces over and under math expressions? What LaTeX markup would one have to write to produce a result similar to the image below?    Please excuse my Paint-and-mouse drawing skills. :-) disconnected phone while installing ROM I recently tried to install a custom ROM in my samsung j6+ but halfway through it my PC turned off. Then the problem starts i decided to reboot my phone but it shows this error &quot;No command&quot; &quot; #Reboot recovery cause is [[check_bootmode]RebootRecoveryWithKey]# Block-based OTA Supported API: 3 # MANUAL MODE v1.0.0# E:Failed to mount / cache: Invalid argument  The error message keeps appearing for just a fraction of a second then disappear and appear again I tried to go to recovery mode but no luck i cant get to it I drained the battery to turn off the device then go download mode but after get to it, it also keep restarting help me please. thank you Microbot Alpha says "All of your minions are considered Microbots."  Does this mean while this card is in play, every single minion in play and in discard and in my hand are considered Microbots?   For every Microbot Fixer in play, does every single minion I have in play now get +1 because they are all microbots? If Microbot Archive is in play, does every single minion I have now cause me to draw a card when destroyed? If I play Microbot Reclaimer, can I shuffle any minions from my discard pile into my deck (by considering them all microbots)?   It looks like this link addresses the first item above:   And it looks like this link addresses the 3rd item, implying that minions in the discard are not minions because they are out of play: Getting MTP enabled devices to work with Ubuntu? Is it a good concept to use multiple inheritance or can I do other things instead? How do you add CSS rules (eg strong { color: red }) by use of Javascript? Debian : Verify disk encryption password for already mounted system I am working on our Debian server on which we have a Raid drive encrypted with 2 devices. I did this around 1.5 years back and have never rebooted since. Now, we want to reboot for some reasons, but I have to make sure that the password I have is the correct one, or we will be stuck.   How can we verify the password for an encrypted-raid? Thank you.  Raid details :    mdadm --detail /dev/md0    /dev/md0:         Version : 1.2   Creation Time : Thu Feb 11 14:43:40 2016      Raid Level : raid1      Array Size : 1953382336 (1862.89 GiB 2000.26 GB)   Used Dev Size : 1953382336 (1862.89 GiB 2000.26 GB)    Raid Devices : 2   Total Devices : 2     Persistence : Superblock is persistent      Update Time : Fri Mar 10 10:37:42 2017           State : clean   Active Devices : 2 Working Devices : 2  Failed Devices : 0   Spare Devices : 0             Name : HOSTNAME:0  (local to host HOSTNAME)            UUID : 5c450558:44a2b1e9:83cb4361:9c74df8e          Events : 49      Number   Major   Minor   RaidDevice State        0       8        1        0      active sync   /dev/sda1        1       8       17        1      active sync   /dev/sdb1   Thank you.   Update  dmsetup table --showkeys CryptVol: 0 Large_Number crypt aes-cbc-essiv:sha256 RANDOM_TEXT  0 253:0 4096 RaidVG-LVMVol: 0 Large_Number linear 9:0 2048 Books automatically deletes my books when offline, and when there is enough space on device Significant figures vs. absolute error On , $N_A = 6.022\;141\;29 \times 10^{23} \text{ mol}^{-1}$ has 9 significant figures and a standard uncertainty of $0.000\;000\;27 \times 10^{23} \text{ mol}^{-1}$.  First, can it be written as the following?  $$ N_A = 6.022\;141\;29 \times 10^{23} \pm 0.000\;000\;27 \times 10^{23} \text{ mol}^{-1} $$ If yes, consider the following quote from  (emphasis added):     Whenever you make a measurement, the number of meaningful digits that you write down implies the error in the measurement. For example if you say that the length of an object is 0.428 m, you imply an uncertainty of about 0.001 m. To record this measurement as either 0.4 or 0.42819667 would imply that you only know it to 0.1 m in the first case or to 0.00000001 m in the second. You should only report as many significant figures as are consistent with the estimated error. The quantity 0.428 m is said to have three significant figures, that is, three digits that make sense in terms of the measurement. Notice that this has nothing to do with the "number of decimal places". The same measurement in centimeters would be 42.8 cm and still be a three significant figure number. The accepted convention is that only one uncertain digit is to be reported for a measurement. In the example if the estimated error is 0.02 m you would report a result of 0.43 ± 0.02 m, not 0.428 ± 0.02 m.   and the following quote from  (emphasis added):     The absolute error places bounds telling you that the actual value is expected to fall between x - D x and x + D x. The absolute error is an easy way to report the error and makes it easy to check that you are not reporting too many significant figures. For example, suppose you had a measurement that was reported to be 10.02 ± 0.3 m. Because the reported value is uncertain in the tenths place, it is wrong to report it to the hundredths place. It should be reported to the same decimal place as the absolute error: 10.0 ± 0.3 m. Another point is that the absolute error and the reported value must have the same units.   If there is such a convention (cf. Rice's lab guide) and it is wrong to report more significant figures than is warranted by the absolute error (cf. the joint project's guidelines), and NIST should have adhered to such convention and striven to be correct, is it true that NIST adheres and is correct because the standard uncertainty has two least significant digits (i.e., $27$)?  Had the standard uncertainty been $0.000\;000\;2\text{ mol}^{-1}$, would NIST have reported Avogadro's Number as $N_A = 6.022\;141\;3\times 10^{23}$ with 8 significant figures? I am using an external application to enter data into a Google Spreadsheet.  This workbook is then referenced by a separate workbook (with the IMPORTRANGE(spreadsheet_key, range_string) function) that manipulates and interprets the data.  My problem is that whenever I personally modify the referenced workbook, the referencing workbook updates just fine, but whenever the external application modifies it, it does not get updated.  I have tried editing the "Spreadsheet settings" so that recalculation is done every minute and on every change.  Furthermore, I have also installed an extension in Google Chrome that auto-refreshes the page every hour.  However, the data does not get re-imported from the referenced workbook.  Even if I copy the formula into a new cell, the data is still not re-imported.  Is there anything that I can do to the referencing workbook to trigger Google Sheets to re-import the data?  Edit: Just to be clear, I currently have one workbook with data entered by an external application in it (call it the "source sheet"), and another workbook with an IMPORTRANGE function in it (call it the "referencing sheet").  The data shown by the IMPORTRANGE function in the "referencing sheet" does not include any data entered by the external application since I last personally edited the "source sheet".  Also, both workbooks are using the new Google Sheets.  Edit: Also, this question is not the same as  because I am using the function given as the solution to that question to import data from a spreadsheet.  The problem is not how to import the data, but rather how to update the source for the data.  My assumption would be that Google would have taken care of this for me, but the data in the "referencing sheet" does not get updated, and the only way I can find to get it to update is to physically go into the "source sheet" and edit it myself. SSL certificate for a public IP address? I just tried buying a Comodo Positive SSL but it was rejected due to not supporting a public IP address, but instead they only support a domain name.  Does anyone know any SSL certificate provider that supports public IP address instead of a domain name?  My company has a dedicated server hosted with a web hosting company, which is used for running a bugtracker for multiple projects (for multiple clients). Since it's only used for a bugtracker, we don't need a domain name (our clients access it by typing the public IP in their browser). Why is there this strange contradiction between the language of logic and that of set theory? In standard probability theory events are represented by sets consisting of elementary events. Consider two events for which (as sets) $A \subset B$. If an elementary event $x \in A$ takes places then we say that $B$ takes place as well. Since $A \subset B$, $x \in A$ implies that $B$ takes place too.  It seems that when for sets (representing events) $$A \subset B$$ $$then \ A \ implies \ B .$$  On the other hand, if we use the language of logic for the events then $$A \supset B$$ $$ \ means\  that \ A\  implies\ B .$$  Why is this strange virtual contradiction between the language of sets and the language of logic?  (In order to avoid down votes and unplesant comments I reveal that I happen to know that the true translation of the sentence $A \supset B$ of logic to the language of sets (representing events) is $\overline{A \cap \overline B}$.) A novel where a woman from "future" of 1999 speaks leet-speak I'm trying to remember the name of a novel/series I read back in the mid-eighties.  I can't remember the author other than that it was a known older author.  I initially thought maybe Saberhagen or Harrison, but can't find it in their bibliographies.  It involved a collection of 4-5 people from different times/places.  The distinctive bit I can remember was that one of them (not the main character) was a cyberpunkish woman from the "future" of 1999 who spoke in futuristic dialect of English that the other characters, including the protagonist, from the "present" (from the eighties) could not understand.  Unfortunately, I don't remember much else of the plot other than that they were making their way through ruins. How did Sauron remain in control of the Nazgûl? The One Ring gave Sauron control of the Nine, yet when it was taken from him he still possessed the power to control them (or, at least he regained it towards the end of the third age).  Why is this?   mentions that he drew the Nine to him, but not how. Display number with leading zeros Given: a = 1 b = 10 c = 100  How do I display a leading zero for all numbers with less than two digits? This is the output I'm expecting: 01 10 100