The Monty Hall Problem and The Birthday Problem

The Monty Hall Problem

The Monty Hall problem is a probability puzzle loosely based on the American television game show Let's Make a Deal and named after the show's original host, Monty Hall. The problem, also called the Monty Hall paradox, is a veridical paradox because the result appears odd but is demonstrably true. The Monty Hall problem, in its usual interpretation, is mathematically equivalent to the earlier Three Prisoners problem, and both bear some similarity to the much older Bertrand's box paradox.

The problem was originally posed in a letter by Steve Selvin to the American Statistician in 1975. (Selvin 1975a) (Selvin 1975b) A well-known statement of the problem was published in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (vos Savant 1990):

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1 [but the door is not opened], and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

Vos Savant's response was that the contestant should always switch to the other door. If the car is initially equally likely to be behind each door, a player who picks Door 1 and doesn't switch has a 1 in 3 chance of winning the car while a player who picks Door 1 and does switch has a 2 in 3 chance. The host has removed an incorrect option from the unchosen doors, so contestants who switch double their chances of winning the car.

Many readers refused to believe that switching is beneficial. After the Monty Hall problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine claiming that vos Savant was wrong. Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy. The Monty Hall problem has attracted academic interest because the result is surprising and the problem is interesting to formulate.

The Birthday Problem

In probability theory, the birthday problem or birthday paradox pertains to the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday (month and day). By the pigeonhole principle, the probability reaches 100% when the number of people reaches 366 (every day of the year). However, 99% probability is reached with just 57 people, and 50% probability with 23 people. These conclusions are based on the assumption that each day of the year (except February 29) is equally probable for a birthday. This is to say, if you take just 57 people you are 99% likely to find two people with same birthday, contrary to the common conclusion it would take near 365 to reach that likelihood.

Note From Josh
See the Wikipedia links at the bottom for the mathematical explanation of these problems. You can also see visual descriptions for each, respectively, here: http://www.youtube.com/watch?v=mhlc7peGlGg and http://www.youtube.com/watch?v=16zXXmSX2v0

From: http://en.wikipedia.org/wiki/Monty_Hall_problem and http://en.wikipedia.org/wiki/Birthday_problem

Castrato

A castrato (Italian, plural: castrati) is a man with a singing voice equivalent to that of a soprano, mezzo-soprano, or contralto voice produced either by castration of the singer before puberty or one who, because of an endocrinological condition, never reaches sexual maturity.

Castration before puberty (or in its early stages) prevents a boy's larynx from being transformed by the normal physiological events of puberty. As a result, the vocal range of prepubescence (shared by both sexes) is largely retained, and the voice develops into adulthood in a unique way. Prepubescent castration for this purpose diminished greatly in the late 18th century and was made illegal in Italy in 1870.

As the castrato's body grew, his lack of testosterone meant that his epiphyses (bone-joints) did not harden in the normal manner. Thus the limbs of the castrati often grew unusually long, as did the bones of their ribs. This, combined with intensive training, gave them unrivalled lung-power and breath capacity. Operating through small, child-sized vocal cords, their voices were also extraordinarily flexible, and quite different from the equivalent adult female voice, as well as higher vocal ranges of the uncastrated adult male. To prevent the child from experiencing the intense pain of castration, many were inadvertently administered lethal doses of opium or some other narcotic, or were killed by overlong compression of the carotid artery in the neck (intended to render them unconscious during the castration procedure).

The training of the boys was rigorous. The regime of one singing school in Rome (c. 1700) consisted of one hour of singing difficult and awkward pieces, one hour practicing trills, one hour practicing ornamented passaggi, one hour of singing exercises in their teacher's presence and in front of a mirror so as to avoid unnecessary movement of the body or facial grimaces, and one hour of literary study; all this, moreover, before lunch. After, half-an-hour would be devoted to musical theory, another to writing counterpoint, an hour copying down the same from dictation, and another hour of literary study. During the remainder of the day, the young castrati had to find time to practice their harpsichord playing, and to compose vocal music, either sacred or secular depending on their inclination. This demanding schedule meant that, if sufficiently talented, they were able to make a debut in their mid-teens with a perfect technique and a voice of a flexibility and power no woman or ordinary male singer could match.

In the 1720s and 1730s, at the height of the craze for these voices, it has been estimated that upwards of 4,000 boys were castrated annually in the service of art. Many came from poor homes and were castrated by their parents in the hope that their child might be successful and lift them from poverty. There are, though, records of some young boys asking to be operated on to preserve their voices

From: http://en.wikipedia.org/wiki/Castrato

Joan Pujol Garcia

Joan Pujol Garcia (14 February 1912 – 10 October 1988), was a double agent during the Second World War who was known by the British codename Garbo and the German codename Arabel. He had a key role in the success of Operation Fortitude, the deception operation intended to mislead the Germans about the timing and location of the invasion of Normandy towards the end of war. The false information Pujol supplied helped persuade German intelligence that the main attack would be in the Pas de Calais, resulting in a decision by the German government to deploy the main body of troops there instead of in Normandy.

Born in the Catalan city of Barcelona, Pujol developed a detestation of Nazi Germany and the Soviet Union after his experience of fascism and communism during the Spanish Civil War. He decided around 1940 that he must make a contribution to the war by helping Britain which, with its Empire, was Germany's only adversary at the time. He initially approached the British but they showed no interest in employing him as a spy. So he resolved to establish himself as a German agent before approaching the British again to offer his services as a double-agent.

Operating initially in Lisbon, he pretended to the Germans that he was in Britain. He fabricated reports about shipping movements based on information gleaned from the library in Lisbon and from newsreel reports he saw in cinemas, and successfully convinced the Germans that he was reporting real information. He claimed to be travelling around Britain and submitted his travel expenses based on fares listed in a British railway guide. During this time he created an extensive network of fictitious sub-agents living in different parts of Britain.

Eventually, he offered his services to British intelligence again. The British had become aware that someone had been feeding the Germans misinformation, and realised the value of misinformation after the German navy wasted resources hunting down a non-existent convoy reported to them by Pujol. This time he was accepted. He was relocated to Britain in the spring of 1942, and operated as a double agent under the aegis of the XX Committee. His spymaster was Cyril Bertram Mills, whom he knew only as 'Mr. Grey'.

On occasion he had to fabricate reasons why his agents had failed to report easily available information that the Germans would eventually know about. For example, he reported that his (fabricated) Liverpool agent had fallen ill just before a major fleet movement from that port on the north-west coast of England. The illness meant that the agent was unable to warn the Germans of the event. To support the story of the illness, the "agent" eventually "died" and a notice was placed in the local newspaper as further evidence to convince the Germans, who were also persuaded to pay a pension to the agent's "widow".

In order to maintain his credibility it was decided that Garbo, or one of his agents, should forewarn the Germans of the timing and some details of the actual invasion of Normandy, although leaving it too late for them to take effective action. Special arrangements were made with the German radio operators to be listening to Garbo through the night of 5/6 June 1944, using the story that a sub-agent was about to arrive with important information. However when the call was made at 3am, no reply was received from the German operators until 8am. Turning this piece of bad luck on its head, Garbo was able to add more details of the operation to the message when finally sent and increase his standing with the Germans.

On June 9 (3 days after D-day), Garbo sent a message to German High Command saying that he had conferred with his agents and developed an order of battle showing 75 divisions in England (when in reality there were only about 50). His message pointed out that units of the First U.S. Army Group (FUSAG) (a fictitious unit which was part of the British deception) had not participated in the invasion and therefore the first landing should be considered a diversion. A German message to Madrid sent two days later said "all reports received in the last week from Arabel [Garbo's German code-name] undertaking have been confirmed without exception and are to be described as exceptionally valuable."

In late June Garbo was instructed by the Germans to report on the falling of V1 flying bombs. Finding no way of giving false information without arousing suspicion, and being unwilling to give correct information, Mills arranged for him to be 'arrested'. He returned to duty a few days later, and forwarded an 'official' letter of apology from the Home Secretary for his unlawful detention.

The Germans paid Garbo (or Arabel, as they called him) US$340,000 to support his network of agents, which at one point totaled 27 fabricated characters.

For his efforts in aid of the Allies Garbo received an MBE from the British; in an ironic twist of fate, following the war he ended up encountering one of his German handlers, who gave him the Iron Cross for his contribution to the German war effort; the Nazis never realized that Garbo had fooled them, and thus he earned the distinction of being one of the few people during World War II to receive decorations from both sides.

After World War II ended, Pujol faked his death and moved to Venezuela, where he lived in anonymity. Mills believed Pujol to be dead, and Pujol had been told that Cyril Mills, or 'Mr. Grey' as he knew him, had been killed as well. However, in 1982 they were emotionally re-united at Mills' home in London. Pujol lived in Lagunillas, Zulia, Venezuela, where he ran a bookstore called 'La Casa del Regalo'. He subsequently moved to Caracas, where he died in 1988. Juan Pujol is buried in Choroní, a town inside Henri Pittier National Park by the Caribbean sea.

Lunokhod Programme

Lunokhod (Russian: Луноход, "Moonwalker") was a series of Soviet robotic lunar rovers designed to land on the Moon between 1969 and 1977. The 1969 Lunokhod 1A was destroyed during launch, the 1970 Lunokhod 1 and the 1973 Lunokhod 2 landed on the moon and the 1977 Lunokhod was never launched. The Lunokhods were primarily designed to support the Soviet manned moon missions and to be used as automatic remote-controlled robots to explore the surface and return pictures. The Lunokhods were transported to the lunar surface by Luna spacecraft, which were launched by Proton rockets. The moon lander part of the Luna spacecraft for Lunokhods were similar to the ones for sample return missions. Not until the 1997 Mars Pathfinder was another remote-controlled vehicle put on an extraterrestrial body.

During its 322 Earth days of operations, Lunokhod 1 traveled 10.5 km and returned more than 20,000 TV images and 206 high-resolution panoramas. In addition, it performed twenty-five soil analyses with its RIFMA x-ray fluorescence spectrometer and used its penetrometer at 500 different locations.

Lunokhod 2 operated for about 4 months, covered 37 km (23 miles) of terrain, including hilly upland areas and rilles, and currently holds the record for the longest distance of surface travel of any extraterrestrial vehicle. It sent back 86 panoramic images and over 80,000 TV pictures. Many mechanical tests of the surface, laser ranging measurements, and other experiments were completed during this time.

For comparison, the similarly sized NASA Mars Exploration Rovers, Spirit and Opportunity had, by their fifth anniversary in January 2009, traveled a total of 21 km (13 mi) and transmitted over 125,000 images.

According to a French documentary TV film "Tank on the Moon" by Jean Afanassieff, the Lunokhod design returned to limelight 15 years later due to the Chernobyl nuclear power plant disaster. The East German made remote controlled bulldozers available to Soviet civil defense troops weighed dozens of tons. Too heavy to operate on the remaining parts of the partially collapsed reactor building roof. Human laborers could not be employed effectively to shovel debris, since work shifts were limited to 90 second intervals due to intense ionizing radiation.

Lunokhod designers were called back from retirement, and in two weeks rovers used nuclear decay heat sources for internal rack climate control, their electronic systems were already hardened to resist radiation. This benefit allowed the 1986 designers to quickly devise a derived vehicle type for nuclear disaster recovery work. Eventually two rovers were delivered to the Chernobyl accident zone and proved useful for clearing debris, earning awards for the designers. Due to extremely high radiation levels, all rovers eventually failed, and human workers (later named liquidators) were called in.

From: http://en.wikipedia.org/wiki/Lunokhod_programme