text
stringlengths 1
2.55k
|
|---|
= = = Maico Buncio = = =
|
Maico Buncio (born as Maico Greg T. Buncio : 10 September 1988 – 15 May 2011) was a Filipino motorcycle racer and four-time Philippine national superbike champion. He died after a race accident in Clark, Pampanga Philippines on 15 May 2011.
|
From Mandaluyong City, Buncio is the son of Gregorio "Yoyong" Buncio, a motorcycle racer, mechanic, and modifier. His mother is Mylene Buncio. He has four other siblings, namely Lourdes, Shara, Jacquelyn, and Barny.
|
Buncio's father gave him the name "Maico" from a European motorcycle brand. At the age of three, he had started to learn riding motorcycles before bicycles. His training began early under the mentoring of his experienced father.
|
Buncio started his motocross competition career at the age of eight. He won first place in the 1996 FBO Motorcross Series 50cc category for ten years old and below held at the TRAKSNIJAK Race Track in Tagaytay City.
|
At the age of 14, Buncio represented the Philippines in Perris, California at the FMF Memorial Day Motorcross Races, winning first place in the Mini Class 85cc for the 14 years old and below category.
|
Buncio took up Bachelor of Science in Commerce, Major in Entrepreneurship at the University of Santo Tomas. Buncio was a consistent honor student since elementary and studied in OB Montessori until high school. He also finished his computer New Media Design Course from Phoenix One Knowledge Solutions in Makati on his latter years.
|
Buncio was the sole representative of the Philippines in the 2004 Yamaha ASEAN Cup held at Shah Alam, Malaysia, where he raced against competitors from Malaysia, Singapore, Thailand, and Indonesia. He won the fourth overall title for the country despite having to start from the back of the grid and having to race against five representing riders each from the other competing countries. This feat earned him the title Rookie of the Year from the organizers of said event.
|
Yamaha rider Buncio dominated the underbone racing 150 cc category for four years since 2006 before placing second to Suzuki rider Johnlery Enriquez in the 2010 event sponsored by Motorcycle Taipei Research Team and PAGCOR Sports.
|
Buncio also broke records at the 3.2-kilometer Batangas Racing Circuit with a lap time of one minute, 49 seconds (400cc Superbike category) and two minutes, seven seconds (Underbone category).
|
2007 was the "most memorable" for Buncio when, at the age of 19, he broke the winning streak of ten-time Rider of the Year, Jolet Jao in the 2007 Shell Advance Superbikes Series. Buncio held the Superbike National Champion title for the next three years.
|
He also finished a race in the AMA Superbike 600cc class in Laguna Seca U.S.A. on September 2008.
|
In 2009, Buncio was chosen as the endorser for Accel Sports Sporteum Philippines Inc.
|
Buncio was also awarded the Golden Wheel Awards Driver of the Year in 2010 and 2011.
|
He has ridden mostly Yamaha bikes in his entire career with Factory Yamaha and YRS until he signed for Team Suzuki Pilipinas in late 2010
|
Maico was also a businessman, opened his own YRS Motorcycle Racing shop in Caloocan, and was a race director of the Moto ROC underbone race series. He is a big fan of Valentino Rossi of MotoGP and always dreamed to compete in international motorcycle racing.
|
He often carries his trademark logo and the iconic race number "129" usually in a form of a sticker on his apparel and motorcycles.
|
Maico Buncio fell a high speed accident on 15 May 2011 during the Superbike qualifying race at the Clark International Speedway Racing Circuit. While passing a semi-straight right hand sweeper on the speedway, Buncio's Suzuki GSX-R 600 motorbike slid and crashed into the run-off section. He was thrown off his bike and landed on an unfinished barrier in the race track. Buncio crashed onto a protruding steel rod of the barrier, which punctured his internal organs, fatally damaging his kidney and liver.
|
Buncio was thrown off some 100 meters from his bike and was impaled on a protruding reed bar in an unfinished barrier on the Clark Speedway Circuit. He was rushed to a nearby hospital in Mabalacat and transferred to the UST Hospital in the wee hours of 15 May and was pronounced dead at 3:57 PM on the same day due to massive internal bleeding. The father of the late superbike champion Maico Buncio raised questions over the first aid procedure done on the rider during his fatal crash at the Clark Speedway Circuit.
|
He said that the Aeromed's medical response team pulled the victim's body from the steel bar. This, he said, might have caused his son's death. The bar punctured his body and damaged his kidney and liver. Instead of cutting the steel bar to free Maico, the AeroMed emergency staffers decided to pull the young motorcycle champion free from his entanglement. This, they learned later, caused massive internal damage to Maico's organs and prompted him to scream in pain.
|
Dr. Reynante Mirano, chief of St. Luke’s Hospital Emergency Medicine, said that instead of pulling Buncio’s body from the protruding metal, the medics should have cut the steel bar.
|
Maico Buncio left a memorable quote to his fellow riders "Never stop riding, because I didn't."
|
Buncio's wake was held in the Loyola Memorial Chapels in Makati City and he was laid to rest in the Loyola Memorial Park in Marikina City on 21 May 2011. A motorcade organized by his fellow riders marked his funeral procession.
|
= = = Maurizio Bianchi = = =
|
Maurizio Bianchi (born 4 December 1955 in Pomponesco in the Province of Mantua) is an Italian pioneer of industrial music, originating from Milan.
|
Bianchi was inspired by the music of Tangerine Dream, Conrad Schnitzler and Throbbing Gristle. He wrote about music for Italian magazines before beginning to release his own cassettes under the name of Sacher-Pelz in August 1979. He released four cassettes as Sacher-Pelz before switching to his own name or simply "MB" in 1980.
|
Bianchi corresponded with many of the key players in the industrial music and noise music scenes including Merzbow, GX Jupitter-Larsen, SPK, Nigel Ayers of Nocturnal Emissions and William Bennett of Whitehouse. After this exchange of letters and music, his first LPs were released in 1981.
|
"Symphony For A Genocide" was released on Nigel Ayers' Sterile Records label after Bianchi had sent Ayers the money to press it. Each track on the LP was named after a Nazi extermination camp. The cover featured photographs of the Auschwitz Orchestra, a group of concentration camp prisoners who were forced to play classical music as people were herded into the gas chambers. The back cover included the text "The moral of this work: the past punishment is the inevitable blindness of the present".
|
Also in 1981, William Bennett, head of the band Whitehouse and the British Come Org. label, offered Bianchi a record contract, which Bianchi signed unchecked. It was based on a "joke contract" that Steven Stapleton of Nurse With Wound had sketched. The contract assumed all rights to Bianchi's work. After delivery of the tapes Bennett edited-in speeches by Nazi leaders, and instead of the relatively unsensational name MB, it was published under the alias "Leibstandarte SS MB", named after the SS unit that worked as bodyguards to Adolf Hitler.
|
By 1983 and the release of the "Plain Truth" LP on U.K. power electronics label Broken Flag, Bianchi had become a Jehovah's Witness. At the end of 1983 Bianchi announced his withdrawal from music, stating "The end is very near, and we have a very short time to recognise our mistakes and to redeem ourselves... I stopped doing music, and now my life is going towards its full awareness".
|
In 1998, encouraged by Alga Marghen label head Emanuele Carcano, who offered him a label of his own, Maurizio Bianchi resumed making music. The label was EEs'T Records, through which he released new editions of old MB albums and many new recordings.
|
Bianchi then proceeded to work on over a hundred new projects both solo or in collaboration with other Italian and international artists including Atrax Morgue, Aube, Francisco López, Mauthausen Orchestra, Merzbow, Ryan Martin and Philip Julian/Cheapmachines.
|
Bianchi has worked with record labels including Dais Records, the Carrboro, North Carolina based Hot Releases and the Italian Menstrual Recordings to re-release some of his out-of-print material.
|
On August 19, 2009, for unspecified personal reasons, Maurizio Bianchi decided again to completely stop making music. This decision was soon after reversed; Maurizio Bianchi continued to release new music.
|
In 2005, a 2-CD-Set named "Blut und Nebel" was released, consisting of a remix of his first ten LPs. Bianchi submitted the set's first CD, remixing the first 5 LPs from 1981 and 1982, to Wikipedia. The track, over 45 minutes long, is split into three .ogg files:
|
Sacher-Pelz
|
MB / Maurizio Bianchi first phase
|
Cassettes
|
Vinyl albums
|
Leibstandarte SS MB
|
As his releases on Come Org have been massively manipulated, Maurizio Bianchi does not count these records as part of his discography. However, in 2013 Triumph of the Will and Weltanschauung were re-issued with bonus tracks as separate CDs and as part of the Teban Slide Art box set, which also contained the unofficial release Lebensraum, all under the name "M.B."
|
MB / Maurizio Bianchi second phase
|
Collaborations
|
= = = FIFA Order of Merit = = =
|
The FIFA Order of Merit is the highest honour awarded by FIFA. The award is presented at the annual FIFA congress. It is normally awarded to people who are considered to have made a significant contribution to :association football.
|
At FIFA's centennial congress they made one award for every decade of their existence. These awards were also handed out to fans, organisations, clubs, and one to African Football. These were referred to as the FIFA Centennial Order of Merit.
|
The winner doesn't have to be directly involved with football to receive it. One such notable non-footballing personality was Nelson Mandela who won it for bringing South Africa back to international football.
|
= = = Fadeaway = = =
|
A fadeaway or fall-away in basketball is a jump shot taken while jumping backwards, away from the basket. The goal is to create space between the shooter and the defender, making the shot much harder to block.
|
The shooter must have very good accuracy (much higher than when releasing a regular jump shot) and must use more strength (to counteract the backwards momentum) in a relatively short amount of time. Also, because the movement is away from the basket, the shooter has less chance to grab his own rebound.
|
The shooting percentage is lower in fadeaway (because of the difficulty of the shot) and the shooter cannot get his own rebound. This leads many coaches and players to believe it is one of the worst shots in the game to take. However, once mastered, it is one of the hardest methods of shooting for defenders to block. The threat of a fadeaway forces a defender to jump into the shooter, and with a pump fake, the shooter can easily get a foul on the defender.
|
Only a handful of great NBA players have been successful shooting fadeaways. Michael Jordan was one of the most popular shooters of the fadeaway. Wilt Chamberlain, Patrick Ewing, LeBron James, Kobe Bryant, Hakeem Olajuwon, Dwyane Wade, Karl Malone, Larry Bird, Carmelo Anthony, DeMar DeRozan, and LaMarcus Aldridge are also well known for using this move. The even more difficult one-legged fadeaway has become Dirk Nowitzki's signature move and has been called by LeBron James the second most unstoppable move ever, only behind Kareem Abdul-Jabbar's skyhook.
|
= = = Two envelopes problem = = =
|
The two envelopes problem, also known as the exchange paradox, is a brain teaser, puzzle, or paradox in logic, probability, and recreational mathematics. It is of special interest in decision theory, and for the Bayesian interpretation of probability theory. Historically, it arose as a variant of the necktie paradox.
|
The problem typically is introduced by formulating a hypothetical challenge of the following type:
|
It seems obvious that there is no point in switching envelopes as the situation is symmetric. However, because you stand to gain twice as much money if you switch while risking only a loss of half of what you currently have, it is possible to argue that it is more beneficial to switch. The problem is to show what is wrong with this argument.
|
Basic setup: You are given two indistinguishable envelopes, each of which contains a positive sum of money. One envelope contains twice as much as the other. You may pick one envelope and keep whatever amount it contains. You pick one envelope at random but before you open it you are given the chance to take the other envelope instead.
|
The switching argument: Now suppose you reason as follows:
|
The puzzle: "The puzzle is to find the flaw in the very compelling line of reasoning above." This includes determining exactly "why" and under "what conditions" that step is not correct, in order to be sure not to make this mistake in a more complicated situation where the misstep may not be so obvious. In short, the problem is to solve the paradox. Thus, in particular, the puzzle is "not" solved by the very simple task of finding another way to calculate the probabilities that does not lead to a contradiction.
|
Many solutions resolving the paradox have been presented. The probability theory underlying the problem is well understood, and any apparent paradox is generally due to treating what is actually a conditional probability as an unconditional probability. A large variety of similar formulations of the paradox are possible, and have resulted in a voluminous literature on the subject.
|
Versions of the problem have continued to spark interest in the fields of philosophy and game theory.
|
The total amount in both envelopes is a constant formula_1, with formula_2 in one envelope and formula_3 in the other.If you select the envelope with formula_2 first you gain the amount formula_2 by swapping. If you select the envelope with formula_3 first you lose the amount formula_2 by swapping. So you gain on average formula_8 by swapping.Swapping is not better than keeping. The expected value formula_9 is the same for both the envelopes. Thus no contradiction exists.
|
The step 7 assumes that the second choice is independent of the first choice. This is the error and this is the source of the apparent paradox.
|
A common way to resolve the paradox, both in popular literature and part of the academic literature, especially in philosophy, is to assume that the 'A' in step 7 is intended to be the expected value in envelope A and that we intended to write down a formula for the expected value in envelope B.
|
Step 7 states that the expected value in B = 1/2( 2A + A/2 )
|
It is pointed out that the 'A' in the first part of the formula is the expected value, given that envelope A contains less than envelope B, but the 'A', in the second part of the formula is the expected value in A, given that envelope A contains more than envelope B. The flaw in the argument is that same symbol is used with two different meanings in both parts of the same calculation but is assumed to have the same value in both cases.
|
A correct calculation would be:
|
If we then take the sum in one envelope to be x and the sum in the other to be 2x the expected value calculations becomes:
|
which is equal to the expected sum in A.
|
In non-technical language, what goes wrong (see Necktie paradox) is that, in the scenario provided, the mathematics use relative values of A and B (that is, it assumes that one would gain more money if A is less than B than one would lose if the opposite were true). However, the two values of money are fixed (one envelope contains, say, $20 and the other $40). If the values of the envelopes are restated as "x" and 2"x", it's much easier to see that, if A were greater, one would lose "x" by switching and, if B were greater, one would gain "x" by switching. One does not actually gain a greater amount of money by switching because the total "T" of A and B (3"x") remains the same, and the difference "x" is fixed to "T/3".
|
Line 7 should have been worked out more carefully as follows:
|
A will be larger when A is larger than B, than when it is smaller than B. So its average values (expectation values) in those two cases are different. And the average value of A is not the same as A itself, anyway. Two mistakes are being made: the writer forgot he was taking expectation values, and he forgot he was taking expectation values under two different conditions.
|
It would have been easier to compute E(B) directly. Denoting the lower of the two amounts by "x", and taking it to be fixed (even if unknown) we find that
|
We learn that 1.5"x" is the expected value of the amount in Envelope B. By the same calculation it is also the expected value of the amount in Envelope A. They are the same hence there is no reason to prefer one envelope to the other. This conclusion was, of course, obvious in advance; the point is that we identified the false step in the argument for switching by explaining exactly where the calculation being made there went off the rails.
|
We could also continue from the correct but difficult to interpret result of the development in line 7:
|
so (of course) different routes to calculate the same thing all give the same answer.
|
Tsikogiannopoulos (2012) presented a different way to do these calculations. Of course, it is by definition correct to assign equal probabilities to the events that the other envelope contains double or half that amount in envelope A. So the "switching argument" is correct up to step 6. Given that the player's envelope contains the amount A, he differentiates the actual situation in two different games: The first game would be played with the amounts (A, 2A) and the second game with the amounts (A/2, A). Only one of them is actually played but we don't know which one. These two games need to be treated differently. If the player wants to compute his/her expected return (profit or loss) in case of exchange, he/she should weigh the return derived from each game by the average amount in the two envelopes in that particular game. In the first case the profit would be A with an average amount of 3A/2, whereas in the second case the loss would be A/2 with an average amount of 3A/4. So the formula of the expected return in case of exchange, seen as a proportion of the total amount in the two envelopes, is:
|
This result means yet again that the player has to expect neither profit nor loss by exchanging his/her envelope.
|
We could actually open our envelope before deciding on switching or not and the above formula would still give us the correct expected return. For example, if we opened our envelope and saw that it contained 100 euros then we would set A=100 in the above formula and the expected return in case of switching would be:
|
As pointed out by many authors, the mechanism by which the amounts of the two envelopes are determined is crucial for the decision of the player to switch or not his/her envelope. Suppose that the amounts in the two envelopes A and B were not determined by first fixing contents of two envelopes E1 and E2, and then naming them A and B at random (for instance, by the toss of a fair coin; Nickerson and Falk, 2006). Instead, we start right at the beginning by putting some amount in Envelope A, and then fill B in a way which depends both on chance (the toss of a coin) and on what we put in A. Suppose that first of all the amount "a" in Envelope A is fixed in some way or other, and then the amount in Envelope B is fixed, dependent on what is already in A, according to the outcome of a fair coin. Ιf the coin fell Heads then 2"a" is put in Envelope B, if the coin fell Tails then "a"/2 is put in Envelope B. If the player was aware of this mechanism, and knows that they hold Envelope A, but don't know the outcome of the coin toss, and doesn't know "a", then the switching argument is correct and he/she is recommended to switch envelopes. This version of the problem was introduced by Nalebuff (1988) and is often called the Ali-Baba problem. Notice that there is no need to look in Envelope A in order to decide whether or not to switch.
|
Many more variants of the problem have been introduced. Nickerson and Falk (2006) systematically survey a total of 8.
|
The simple resolution above assumed that the person who invented the argument for switching was trying to calculate the expectation value of the amount in Envelope A, thinking of the two amounts in the envelopes as fixed ("x" and 2"x"). The only uncertainty is which envelope has the smaller amount "x". However, many mathematicians and statisticians interpret the argument as an attempt to calculate the expected amount in Envelope B, given a real or hypothetical amount "A" in Envelope A. (A mathematician would moreover prefer to use the symbol "a" to stand for a possible value, reserving the symbol "A" for a random variable). One does not need to look in the envelope to see how much is in there, in order to do the calculation. If the result of the calculation is an advice to switch envelopes, whatever amount might be in there, then it would appear that one should switch anyway, without looking. In this case, at Steps 6, 7 and 8 of the reasoning, "A" is any fixed possible value of the amount of money in the first envelope.
|
This interpretation of the two envelopes problem appears in the first publications in which the paradox was introduced in its present-day form, Gardner (1989) and Nalebuff (1989). It is common in the more mathematical literature on the problem. It also applies to the modification of the problem (which seems to have started with Nalebuff) in which the owner of Envelope A does actually look in his envelope before deciding whether or not to switch; though Nalebuff does also emphasize that there is no need to have the owner of Envelope A look in his envelope. If he imagines looking in it, and if for any amount which he can imagine being in there, he has an argument to switch, then he will decide to switch anyway. Finally, this interpretation was also the core of earlier versions of the two envelopes problem (Littlewood's, Schrödinger's, and Kraitchik's switching paradoxes); see the concluding section, on history of TEP.
|
This kind of interpretation is often called "Bayesian" because it assumes the writer is also incorporating a prior probability distribution of possible amounts of money in the two envelopes in the switching argument.
|
The simple resolution depended on a particular interpretation of what the writer of the argument is trying to calculate: namely, it assumed he was after the (unconditional) expectation value of what's in Envelope B. In the mathematical literature on Two Envelopes Problem a different interpretation is more common, involving the conditional expectation value (conditional on what might be in Envelope A). To solve this and related interpretations or versions of the problem, most authors use the Bayesian interpretation of probability, which means that probability reasoning is not only applied to truly random events like the random pick of an envelope, but also to our knowledge (or lack of knowledge) about things which are fixed but unknown, like the two amounts originally placed in the two envelopes, before one is picked at random and called "Envelope A". Moreover, according to a long tradition going back at least to Laplace and his principle of insufficient reason one is supposed to assign equal probabilities when one has no knowledge at all concerning the possible values of some quantity. Thus the fact that we are not told anything about how the envelopes are filled can already be converted into probability statements about these amounts. No information means that probabilities are equal.
|
In steps 6 and 7 of the switching argument, the writer imagines that that Envelope A contains a certain amount "a", and then seems to believe that given that information, the other envelope would be equally likely to contain twice or half that amount. That assumption can only be correct, if prior to knowing what was in Envelope A, the writer would have considered the following two pairs of values for both envelopes equally likely: the amounts "a"/2 and "a"; and the amounts "a" and 2"a". (This follows from Bayes' rule in odds form: posterior odds equal prior odds times likelihood ratio). But now we can apply the same reasoning, imagining not "a" but "a/2" in Envelope A. And similarly, for 2"a". And similarly, ad infinitum, repeatedly halving or repeatedly doubling as many times as you like. (Falk and Konold, 1992).
|
Suppose for the sake of argument, we start by imagining an amount 32 in Envelope A. In order that the reasoning in steps 6 and 7 is correct "whatever" amount happened to be in Envelope A, we apparently believe in advance that all the following ten amounts are all equally likely to be the smaller of the two amounts in the two envelopes: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 (equally likely powers of 2: Falk and Konold, 1992). But going to even larger or even smaller amounts, the "equally likely" assumption starts to appear a bit unreasonable. Suppose we stop, just with these ten equally likely possibilities for the smaller amount in the two envelopes. In that case, the reasoning in steps 6 and 7 was entirely correct if envelope A happened to contain any of the amounts 2, 4, ... 512: switching envelopes would give an expected (average) gain of 25%. If envelope A happened to contain the amount 1, then the expected gain is actually 100%. But if it happened to contain the amount 1024, a massive loss of 50% (of a rather large amount) would have been incurred. That only happens once in twenty times, but it is exactly enough to balance the expected gains in the other 19 out of 20 times.
|
Alternatively we do go on ad infinitum but now we are working with a quite ludicrous assumption, implying for instance, that it is infinitely more likely for the amount in envelope A to be smaller than 1, "and" infinitely more likely to be larger than 1024, than between those two values. This is a so-called improper prior distribution: probability calculus breaks down; expectation values are not even defined; see Falk and Konold and (1982).
|
Many authors have also pointed out that if a maximum sum that can be put in the envelope with the smaller amount exists, then it is very easy to see that Step 6 breaks down, since if the player holds more than the maximum sum that can be put into the "smaller" envelope they must hold the envelope containing the larger sum, and are thus certain to lose by switching. This may not occur often, but when it does, the heavy loss the player incurs means that, on average, there is no advantage in switching. Some writers consider that this resolves all practical cases of the problem.
|
But the problem can also be resolved mathematically without assuming a maximum amount. Nalebuff (1989), Christensen and Utts (1992), Falk and Konold (1992), Blachman, Christensen and Utts (1996), Nickerson and Falk (2006), pointed out that if the amounts of money in the two envelopes have any proper probability distribution representing the player's prior beliefs about the amounts of money in the two envelopes, then it is impossible that whatever the amount "A=a" in the first envelope might be, it would be equally likely, according to these prior beliefs, that the second contains "a"/2 or 2"a". Thus step 6 of the argument, which leads to "always switching", is a non-sequitur, also when there is no maximum to the amounts in the envelopes.
|
The first two resolutions discussed above (the "simple resolution" and the "Bayesian resolution") correspond to two possible interpretations of what is going on in step 6 of the argument. They both assume that step 6 indeed is "the bad step". But the description in step 6 is ambiguous. Is the author after the unconditional (overall) expectation value of what is in envelope B (perhaps - conditional on the smaller amount, "x"), or is he after the conditional expectation of what is in envelope B, given any possible amount "a" which might be in envelope A? Thus, there are two main interpretations of the intention of the composer of the paradoxical argument for switching, and two main resolutions.
|
A large literature has developed concerning variants of the problem. The standard assumption about the way the envelopes are set up is that a sum of money is in one envelope, and twice that sum is in another envelope. One of the two envelopes is randomly given to the player ("envelope A"). The originally proposed problem does not make clear exactly how the smaller of the two sums is determined, what values it could possibly take and, in particular, whether there is a minimum or a maximum sum it might contain. However, if we are using the Bayesian interpretation of probability, then we start by expressing our prior beliefs as to the smaller amount in the two envelopes through a probability distribution. Lack of knowledge can also be expressed in terms of probability.
|
A first variant within the Bayesian version is to come up with a proper prior probability distribution of the smaller amount of money in the two envelopes, such that when Step 6 is performed properly, the advice is still to prefer Envelope B, whatever might be in Envelope A. So though the specific calculation performed in step 6 was incorrect (there is no proper prior distribution such that, given what is in the first envelope A, the other envelope is always equally likely to be larger or smaller) a correct calculation, depending on what prior we are using, does lead to the result formula_15 for all possible values of "a".
|
In these cases it can be shown that the expected sum in both envelopes is infinite. There is no gain, on average, in swapping.
|
Though Bayesian probability theory can resolve the first mathematical interpretation of the paradox above, it turns out that examples can be found of proper probability distributions, such that the expected value of the amount in the second envelope given that in the first does exceed the amount in the first, whatever it might be. The first such example was already given by Nalebuff (1989). See also Christensen and Utts (1992).
|
Denote again the amount of money in the first envelope by "A" and that in the second by "B". We think of these as random. Let "X" be the smaller of the two amounts and "Y=2X" be the larger. Notice that once we have fixed a probability distribution for "X" then the joint probability distribution of "A,B" is fixed, since "A,B" = "X,Y" or "Y,X" each with probability 1/2, independently of "X,Y".
|
The "bad step" 6 in the "always switching" argument led us to the finding "E(B|A=a)>a" for all "a", and hence to the recommendation to switch, whether or not we know "a". Now, it turns out that one can quite easily invent proper probability distributions for "X", the smaller of the two amounts of money, such that this bad conclusion is still true. One example is analysed in more detail, in a moment.
|
As mentioned before, it cannot be true that whatever "a", given "A=a", "B" is equally likely to be "a"/2 or 2"a", but it can be true that whatever "a", given "A=a", "B" is larger in expected value than "a".
|
Suppose for example (Broome, 1995) that the envelope with the smaller amount actually contains 2 dollars with probability 2/3 where "n" = 0, 1, 2,… These probabilities sum to 1, hence the distribution is a proper prior (for subjectivists) and a completely decent probability law also for frequentists.
|
Imagine what might be in the first envelope. A sensible strategy would certainly be to swap when the first envelope contains 1, as the other must then contain 2. Suppose on the other hand the first envelope contains 2. In that case there are two possibilities: the envelope pair in front of us is either {1, 2} or {2, 4}. All other pairs are impossible. The conditional probability that we are dealing with the {1, 2} pair, given that the first envelope contains 2, is
|
End of preview. Expand
in Data Studio
- Downloads last month
- 5