Abstract The authorities' version of events on September 11, 2001, is riddled with ludicrous absurdities, suspension of the laws of physics, and vanishingly unlikely coincidences. Although the mass media has gone along with the official view of Muslim kamikaze pilots/hijackers, the ongoing Internet expose of scores of problems with the suicide pilots scenario has led to widespread recognition that it is false. Previous analyses have tended to be qualitative rather than quantitative. By examining events whose truth-value is closely correlated with that of a particular theory and calculating the improbability of a false match with alternative theories, we shall show the most likely apportionment of culpability, how the conspiracy evolved, and what actually happened on the day. The available evidence is more than sufficient to convict the principals in a criminal court of law. We find governments not guilty of conspiracy to mass murder their own citizens, but guilty of conspiracy to wage illegal wars of aggression, conspiracy to mass murder and maim unspecified foreign persons, conspiracy to sacrifice their own military for profit, conspiracy to wilfully subject hundreds of their own civilians to extreme danger to life and limb, culpable negligence in failing to protect thousands of their civilians from death and injury, and accessory to mass murder. The Bush Administration was directly responsible for the Pentagon incident, the state of Israel approved and organised the flying of planes into the WTC, and a powerful real estate consortium arranged the demolition of three skyscrapers. The recommendation is not that members of some specific race be persecuted or exterminated, but that Lord Acton's famous maxim is so important that civilisation is unlikely to endure short of a wholesale reform of traditional institutions and organisation. Methodology We select two theories to weigh against each other, where we suspect that Theory F is false and Theory T is true. It is assumed that Theories F and T cannot both be true, although they could both be false. Specifically, we are considering the truth or falsehood - the truth-value - of alternative conflicting theories concerning the identities of the perpetrators of 9/11. There will exist a particular set of events whose truth-value is very closely correlated with that of a selected theory. For example, if the official account of 9/11/01 events is true, then it is associated with a set of events which are an integral part of the theory and hence explicable. If an alternative theory is true, then another set of events are likely to be true, having been generated by a causal process inherent to the latter theory. Either way, events with a unique causal link to the true theory are likely to occur, and if that theory were false then those events could only occur as a result of a random generating process and so would be very improbable. Events must satisfy two conditions to quality for inclusion in the set with a very closely correlated truth-value: (i) They are likely to have occurred if Theory T is true (e.g. P greater than 0.1). (ii) They are very unlikely to have occurred if Theory T is false (e.g. P less than 0.001), irrespective of whether Theory F is true or false. Let Theory F be the official version of events "Bin Laden Incited a Rabid Satanism" (BLIARS), and let Theory T be "Operation Snowball" - the true events of 9/11. (Once the conspiracy started rolling, it grew to a magnitude totally out of proportion to the original plot. And those who signed up to the original, limited casualties scheme hadn't a snowball in hell's chance of turning back the tide.) Asking the question, "Did a flying object hit the Pentagon on 9/11?" would not be useful, since it fails to distinguish between alternative theories as to the perpetrators. The question, "Did Flight 77, a Boeing 757, hit the Pentagon on 9/11?" is useful, since this does distinguish between theories. The ideal type of question would both distinguish between theories and feature an undeniable event, such as, "Did three steel-framed skyscrapers collapse on 9/11?" For a series of preselected independent events, the probability for all events to occur is equal to the product of the individual probabilities for each event to occur. However, when we become aware of a number of unlikely occurrences after the event, these events have necessarily been selected from a larger set of world events, many of which may be quite normal. Thus, in such a case, the probability of the series may be considerably greater than the product of the individual probabilities. The improbability will be diluted to a degree which is determined by the size of the larger set, and by the number of improbable events under analysis. Acausal coincidences are regularly reported, such as X answering a ringing telephone booth and finding Y is trying to phone X but dialled the wrong number. Intuitively, such events can seem more frequent than chance would dictate. In fact, the set of events is so vast that such improbable events are included and their occurrence is consistent with a random generating process. But we are concerned with a specific set of events that relate to the identities of the 9/11 perpetrators. Since the truth-value of each event must have a sufficiently high degree of positive correlation with that of Operation Snowball, the set of events is very limited. The argument approaches a reductio ad absurdum, if we consider Operation Snowball as the conclusion and the set of positively correlated events as the premises. When we assume BLIARS to be true and Snowball false, the series of Snowball-correlated events approaches but stops short of impossibility. In a criminal investigation, guilt can never be proven to the same degree of certainty as our knowledge that two and two is four or no mammals are cold-blooded. Witnesses can lie; 'evidence' can be planted. But, as we shall calculate below, given the limited size of the set into which dilution of the historical improbable series has occurred, the series can remain sufficiently improbable to furnish a reliable criminal conviction. When BLIARS is assumed to be false and Snowball true, the series of events turns from almost impossible to very reasonable. (The events within the series will only have dependent probabilities if the truth-values of Snowball and BLIARS are in a floating state. When we define one or the other as true then the probabilities are independent and they may be multiplied prior to the necessary correction for being selected after the event.) In the case of 9/11, the guilt of the principal perpetrators can be proven to at least the same degree of certainty as a case involving DNA profiling and multiple corroborative evidence. Calculating the corrected probability In order to demonstrate how the improbability of a series of improbable events is diluted as elements are added to the set containing the series, it will be useful to consider random number generators with integer outputs from 00 to 99, or 000 to 999 etc. The output range corresponds to a sample space of weighted outcomes, and the state of a particular digit denotes the truth-value of a particular event. The findings will indicate whether it is worth taking into account dozens of anomalies or more useful to concentrate on a few highly aberrant events. Suppose there are two independent events whose probability is 0.1 or 1 in 10. If these were the only two events in the set, the sample space of all possibilities could be represented by the random generation of a two-digit number. "0" denotes an occurrence; "N" denotes anything from 1-9 and non-occurrence; "A" denotes any outcome from 0-9. So occurrence of both events requires a 00 and has 0.01 or 1 in 100 probability as expected. If the set contains three events, the possibilities for at least two probability 0.1 events includes 00N, 0N0, N00 and 000 which is 3 x 9 + 1 = 28 from a total of 1,000 possibilities. With a four-element set, the 10,000 combinations offers 00NN, 0N0N, 0NN0, N00N, N0N0, NN00, 000N, 00N0, 0N00, N000, and 0000 which is 6 x 9 ^ 2 + 4 x 9 ^ 1 + 1 x 9 ^ 0 = 523. A five-element set offers 00NNN, 0N0NN, 0NN0N, 0NNN0, N00NN, N0N0N, N0NN0, NN00N, NN0N0, NNN00, 000NN, 00N0N, 00NN0, 0N00N, 0N0N0, 0NN00, N000N, N00N0, N0N00, NN000, 0000N, 000N0, 00N00, 0N000, N0000, 00000 which totals 10 x 9 ^ 3 + 10 x 9 ^ 2 + 5 x 9 ^ 1 + 1 x 9 ^ 0 = 8,146 from 100,000. A six-element set offers 00NNNN, 0N0NNN, 0NN0NN, 0NNN0N, 0NNNN0, N00NNN, N0N0NN, N0NN0N, N0NNN0, NN00NN, NN0N0N, NN0NN0, NNN00N, NNN0N0, NNNN00, 000NNN, 00N0NN, 00NN0N, 00NNN0, 0N00NN, 0N0N0N, 0N0NN0, 0NN00N, 0NN0N0, 0NNN00, N000NN, N00N0N, N00NN0, N0N00N, N0N0N0, N0NN00, NN000N, NN00N0, NN0N00, NNN000, 0000NN, 000N0N, 000NN0, 00N00N, 00N0N0, 00NN00, 0N000N, 0N00N0, 0N0N00, 0NN000, N0000N, N000N0, N00N00, N0N000, NN0000, 00000N, 0000N0, 000N00, 00N000, 0N0000, N00000, 000000 which totals 15 * 9 ^ 4 + 20 * 9 ^ 3 + 15 * 9 ^ 2 + 6 * 9 ^ 1 + 1 * 9 ^ 0 = 114,265 from 1,000,000. With 4 events from a set of nine, for example, there are 126 combinations of four zeros and five Ns, 126 of five zeros and four Ns, 84 of six zeros and three Ns, 36 of seven zeros and two Ns, 9 of eight zeros and a single N, and one combination of nine zeros. Allowing for the descending powers of nine, this totals 8,331,094 combinations from a sample space of one billion. With 2 improbable events from a set of three, the probability is increased from 0.01 by a factor of 2.8; with 2 from 4 the multiplying factor is 5.23; with 2 from 5 it is 8.146; with 2 from 6 it is 11.4265. In the previous paragraph's example of 4 events from 9, the probability increases by a factor of 83.31094 as a result of the five additional elements. For our purposes, a useful approximation can be obtained from the formula for the number of combinations of n elements taken r at a time, where: n! nCr = ------------- r! * (n - r)! n is the number of elements or events in the set r is the number of selected improbable events within the set. Multiply the unadjusted probability of the series of events by nCr, or divide its reciprocal the unadjusted improbability by nCr, in order to obtain an adjusted measure which allows for the other elements of the set. These elements, consisting solely of events which meet the positive correlation of truth-value qualifications, would include events not conceived of and others that failed to occur. The error introduced by this approximation exaggerates the dilution of improbability with increasing n, which biases in favour of the BLIARS theory by reducing its improbability. For example, allowing the combinations 00A, 0A0 and A00 ("A" being any outcome from 0 to 9) suggests 3 x 10 = 30 from 1,000, but this incorrectly counts 000 three times, The accurate but more complex method above (with the algorithm outlined two paragraphs below) shows the r=2, n=3 case comprising 00N, 0N0, N00 and 000 which is really 28 from 1,000 combinations. However, the error reduces as the number base B of our thought experiment random generator increases. If we concentrate on a few improbable events with probabilities of less than 1 in 10,000, each digit of the random number has over 10,000 possibilities. Provided the number base B remains high, the nCr approximation remains considerably more accurate than our estimate of the probability of each event or the size n of the set. This holds for larger values of r, the number of selected improbable events. For example, suppose we have four events assumed to be of equal probability, and the product of their probabilities is 1 / (3.2 * 10 ^ 25). We set B equal to the fourth root of 3.2 * 10 ^ 25, and find that the approximation is exaggerating the improbability dilution by only some 0.0000336% for every element in the set in excess of four. (0.0000336% being about 80% of the reciprocal of the fourth root of 3.2 * 10 ^ 25,) The more precise calculation is obtained from a series with descending powers of B. In fact, we needn't expand (B - 1). Input the variables B, n and r, and the number of combinations is totalled by summing the number of combinations of each power of (B - 1). Start with (B - 1) ^ (n - r) multiplied by nCr for the actual values of n and r. Then for each subsequent iteration r is incremented by one until r = n. The accumulated total of all iterations yields the total number of combinations. This total is then multiplied by B ^ (r - n) which compensates for the larger sample space, to obtain the actual factor for division of improbability or multiplication of probability. Input B Input n Input r Set grossimprob to B ^ r Set combinations to zero For iteration is r to n a is n Call factorialcalculator nfact is factorial a is iteration Call factorialcalculator rfact is factorial a is n - iteration Call factorialcalculator nmrfact is factorial nCr is nfact / (rfact * nmrfact) combinations is combinations + (B - 1) ^ (n - iteration) * nCr Endfor Set truedilutionfactor to combinations * B ^ (r - n) Print "True dilution factor ="; truedilutionfactor Print "True probability = 1 in"; grossimprob / truedilutionfactor Run factorialcalculator factorial is 1 For integer is 1 TO a factorial is factorial * integer Endfor Return We shall make use of another approximation which allows us to apply either of the above formulae to the complete set of events, eliminating the need to split events of various probability into alternative sections and apply the formulae to each section in turn: Let the product of the individual probabilities of the selected improbable events be P, and the number of selected improbable events be r. Then the probability of each improbable event is assumed to be the rth root of P, and denoted by B. This approximation also biases in favour of BLIARS. Consider an example of two 1 in 100 events and two 1 in 10000 events in a set of ten. If we do the calculation for four 1 in 1000 events in a set of ten, the nCr approximation indicates an improbability dilution of 210, since if n = 10, r = 4, then nCr = 210. The unadjusted probability of 1 in 10 ^ 12 is increased to 1 in 4.7619 * 10 ^ 9. Using the exact algorithm, it is actually 1 in 4.7848 * 10 ^ 9, with the nCr approximation exaggerating the improbability dilution. Now compare with the actual case of two 1 in 100 events in a set of five along with two 1 in 10000 events in another set of five. For the two 1 in 100 events the improbability dilution is 9.801496 by the exact method, and for the two 1 in 10000 events it is 9.99800015. So the additional six elements has raised the probability from 1 in 10 ^ 12 by a factor of nearly 100, with the corrected probability at 1 in 1.020456495 * 10 ^ 10. The nCr approximation indicates a factor of 10 for each set of five, i.e. 100 in all and a corrected probability of 1 in 10 ^ 10. So the equal probabilities (EP) approximation has raised the corrected probability by a factor of more than two, the nCr approximation raised it by a little over 2%, and applying both exaggerated the dilution of improbability by a little more than application of the EP alone. To summarise why the EP approximation exaggerates dilution of improbability, in the above paragraph 210 is greater than 10 squared. For some given values of n and r, a doubling of both will result in nCr increasing to rather more than its square. The critic might consider that if we are going to split into sections for different probabilities, then maybe the higher probability events (being more commonplace) should take up more than half of the total set, and perhaps this would favour the BLIARS theory and show that the EP approximation is not kind to the BLIARS theory after all. It's a sound approach. Let's suppose we have two 1 in 100 events from a set of six and two 1 in 10000 events from a set of four. Firstly, for the two 1 in 100 events from a set of six, the exact algorithm shows that the actual dilution factor is 14.60447605. Then, for the two 1 in 10000 events from a set of four, the exact method shows the dilution factor is 5.99920003. The product of the two is only 87.61517316 compared to 97.99535848 or greater before. An unequal split lowers dilution and probability and raises improbability. Hence, it does not help the BLIARS theory, the equal split EP does instead! So we imagine a random number generator with the number base B being the reciprocal of the rth root of P. A '0' is required to denote the occurrence of an improbable event. Since each digit has over 10,000 variations, 'A' (any outcome) is very nearly equal to 'N' (any outcome but '0'), and the dilution of improbability is obtained fairly accurately from the approximate formula below: The corrected probability for the series of improbable events, after allowing for the extra elements in the set, is given by P * n! Pdiluted = ------------- r! * (n - r)! where P is the product of the individual probabilities of unlikely events n is the number of elements or events in the set r is the number of selected improbable events within the set. or Pdiluted = P * nCr with many calculators carrying the nCr function. To conclude, our approach will analyse a few very low probability events rather than a large quantity of fairly low probability events. The latter would require the more complex algorithm to compute improbability dilution to sufficient precision, identification of additional anomalous events, and in any case would establish guilt to a lower degree of certainty. Given that we assume that the total number n of elements in the set is more than twice the number r of selected improbable events (which we do), it follows that for a given product of probabilities P, any increase in r will increase the dilution of improbability. Also, increasing r to include higher probability events with a lower correlation with the test theory's truth-value implies that we should assume a greater value for n, which would further dilute the improbability. Identification of large numbers of anomalies has already been achieved on numerous websites to provide interesting qualitative analyses. Our approach can identify the guilty parties with a confidence factor approaching certainty, whilst using a readily available algorithm provided in many scientific calculators. The official BLIARS version of events Several years after the event, websites continue to flourish and papers continue to emerge. Some sites detail well over 100 problems with the BLIARS version of events. Fortunately, we need not estimate improbabilities for every aspect of the story. According to BLIARS, on the morning of September 11, 2001, 19 Arab hijackers took over four planes and in three cases successfully navigated and steered the planes into their targets. Three New York skyscrapers collapsed as a result of fires after planes hit two of them, a third plane damaged the Pentagon, and a fourth crashed into the ground at Shanksville, Pennsylvania following a revolt by passengers. The hijackers were armed with box cutters and plastic knives which they had smuggled on board. They had also managed to bring a gun and a bomb on Flight 93, and gas canisters and possibly a gun on Flight 11. According to some reports, they had also smuggled electronic navigation units on board. In all cases, the hijackers managed to take control of the plane without the pilot or co-pilot being able to type in the four-digit piracy distress code to warn ground control of a hijacking, the hijackers managed to switch off the plane's transponder, they managed to avoid being photographed by any airport CCTVs when embarking at Boston, Dulles or Newark, they managed to avoid being listed in any of the passenger manifests, and they managed to evade any defensive response by the mighty US Air Force. (July 2004 update below, regarding "Dulles hijackers video".) The operation was masterminded by Osama Bin Laden, a wealthy, brilliant but evil Muslim terrorist whose ambition appears to be world domination and the elimination of "freedom". Alternative accounts hold that Saddam Hussein was a key conspirator. The authorities said that they "didn't see this one coming", it was "something we had never even thought of", and "there were no warning signs". Nonetheless, within hours, the valiant vigilant FBI knew who did it. Cell phone calls from the doomed aircraft passengers, including Barbara Olson the US solicitor general's wife, provided some of the first hard evidence pointing to Muslim terrorism. By 9 p.m. on the day of the hijackings, police and FBI agents towed a white Mitsubishi Mirage, rented by Mohamed Atta and used by five of the hijackers, which had been found at Boston's Logan Airport containing incriminating evidence such as an Arabic language flight manual. At 11 p.m. that day, a blue Nissan Altima also rented by Atta was found at Portland International Airport, Maine, and towed away four hours later. (ABC News of September 12.) Within a day of the hijackings, the slightly charred passport of hijacker Satam Al Suqami was found in the World Trade Centre rubble. FBI press releases of September 14th and 27th each included the same list of 19 hijackers' names. July 2004 update: We are addressing the official story as of the first half of 2004. If the official story changes almost three years after the fact to include a "surveillance video" that has suddenly emerged, this does not merit a lengthy response. If CCTV had recorded any hijackers at Boston, Washington or Newark, there is no doubt that media outlets would have broadcast images within a matter of days or hours. It took just two to six months to produce the animation for each episode of the highly acclaimed series "Walking with Dinosaurs", so allowing three years to fake a security video is rather like allowing a candidate three years to write in the answers to a two-hour exam, given unlimited access to computers and the answer paper. The grainy "Dulles hijackers surveillance video" lacked time and date stamps, yet the authorities still claimed to know the time of each section, to the minute. At the alleged time of the hijackers' security checks, 07:18 on 9/11 which was half an hour after sunrise at Dulles airport, the solar elevation angle was only 5.1 degrees. At solar noon, 13:06 local time, the elevation angle was 55.4 degrees. Thus, at 07:18 an object would have cast a shadow 11.2 times its height (from 1 / tan [5.1 deg.]), and 16.2 times longer than its 13:06 true noon shadow. The natural light intensity at 07:18 was 9.26 times lower than it was at 13:06 (from sin [5.1 deg.] / sin [55.4 deg.]). Washington weather records also show that there was some cloud around from 06:51 to 07:51. Judging by the video's brilliance of light streaming in through the airport entrance and shortness of shadows cast by cabs, the forgers not only failed to fake the video at the right time of day, they also produced it close to the summer solstice in late June, which would concur with its July broadcasting. October 2004 update: Just four days before the US elections, a video of "Osama Bin Laden" conveniently appeared on TV. This is a man said to be living in a cave in mountains near the Pakistan / Afghanistan border, yet the video's background suggests that the recording was made in a TV studio. US intelligence services claimed to know exactly where he is; they "just can't get him". In that case, why not blockade the area and prevent movement of supplies? "Bin Laden" or the actor playing him admitted responsibility for 9/11 - why wait for over three years? The forgers should have shot the video on location, say somewhere like the Rockies or Appalachian range. Or in a Hollywood studio they could at least have used cardboard cutouts of a mountainous panorama, as in early westerns or children's TV. The BLIARS theory vs Gross Incompetence / Bush Administration Complicity The BLIARS theory itself implies gross incompetence and a massive cover-up. Let's consider some of the problems with BLIARS that might be resolved by substituting an incompetence theory which retains the Muslim kamikaze kernel. The nineteen named hijackers: The problem here is that at least eight of these 19 "suicide hijackers" had turned up alive and well by September 23, 2001, plus a ninth by September 29, having mostly been in Saudi Arabia at the time of the attacks. It didn't stop the FBI from releasing the same 19 names on September 27 that they had already released on the 14th. They would not let facts get in the way of a good story. Authorities had never conceived of planes being used as bombs: Unfortunately for this claim, the FBI and CIA had been aware since 1995 of plans to simultaneously hijack several planes with the options of blowing them up over the Pacific or crashing them into targets in the US such as the World Trade Centre. In the weeks leading up to 9/11, the US government received intelligence from Egypt, France, Iran, Israel, Russia and other sources, warning of an imminent attack. Some of these warnings were specifically about hijacked airplanes to be used against buildings. And the National Reconnaissance Office (NRO), based in Chantilly, Virginia and intrinsically entangled with the CIA and the Department of Defense (DoD), had scheduled an exercise starting at 9:00 on September 11, 2001, involving an aircraft hitting one of its buildings. There were also a number of wargames operating simultaneously, controlled by Dick Cheney: Vigilant Guardian, Vigilant Warrior, Northern Guardian, Northern Vigilance, and Tripod II. The first three were all to do with hijacked airplanes in NE US airspace. Some of these drills were live-fly exercises with remote-controlled planes, simulating the behaviour of hijacked airliners and coinciding with the alleged start of the "Arab hijackings". US Air Force defences stand down: This has been well documented. The world's most powerful air force routinely intercepted commercial or civilian planes that strayed off course as part of its standard operating procedures, whenever the problem could not be resolved through radio contact. From September 2000 to June 2001, for example, jet fighters were scrambled 67 times. Major Douglas Martin, a NORAD spokesman, said that an order must be given by President Bush or Donald Rumsfeld prior to the ultimate response of shooting down a suspect plane. However, the preliminary measures of scrambling, intercepting, attracting the errant pilot's attention by rocking the fighter's wingtips or passing in front of the plane or firing tracer rounds in its path, would require no such order. The BLIARS version of events evolved in the days immediately following 9/11, as the Bush Administration floated various cover stories. On September 13 General Richard Myers claimed that fighter jets had not been scrambled until after the Pentagon was hit. Later revisions held that fighters had in fact been scrambled from Otis ANG Base, Cape Cod, Massachusetts at 8:52, and from Langley AF Base at 9:30, seven minutes before the Pentagon was hit. In these versions, fighter jets were scrambled, but arrived too late because inexplicably they only flew at some 25% of their maximum speed, and the Andrews AF Base which is supposed to maintain a ready response force and is only 10 miles from Washington DC, did not have fighters available. BLIARS maintains that at 8:20 AA Flight 11's transponder was turned off, and at 8:25 two flight attendants had alerted American Airlines, and Boston air traffic control had heard a hijacker saying, "We have some planes. Just stay quiet and you will be OK." Since it was not until about 9:37 that AA 77 struck the Pentagon and UA 93 did not crash near Shanksville until around 10:06, the authorities had over an hour's warning that hijackings were in progress. Yet whichever cover story we assume, the response was too little too late. Widespread scattering of Flight 93 wreckage: Part of the wreckage including engine and body parts was found eight miles from the main crash site, with more debris at a location six miles distant. According to the BLIARS account, debris might have been blown eight miles by the wind. Yet windspeed was only around 10mph on the day. A light breeze is hardly likely to blow engine parts and body parts up out of a crater and over a distance of eight miles. And if so, there would be a continuous debris field, rather than debris concentrated at three main sites. Eyewitnesses heard explosions, saw a fireball and white smoke coming from the plane before the impact, and saw a second white jet nearby when the Boeing 757 came down. Thus, the evidence is indicative of the plane having been shot down, and inconsistent with the story of the "passengers' revolt" - which had shades of the Jessica Lynch POW "plucky damsel rescued from sadistic Iraqis" myth. Cell phone calls: These are extremely unreliable at altitudes and velocities anywhere near to the typical Boeing 757 and 767 cruising altitude of 35000 feet or cruising speed of 530 mph. At high altitudes the signal is attenuated such that a successful connection is highly improbable. It is now claimed that more than 30 phone calls were made from Flight 93, lasting up to 26 minutes, and mostly through mobile cell phones. Why would terrorists allow passengers to make calls which might jeopardise the mission? With four terrorists on UA 93, one (Jarrah) remained in the cockpit leaving three to subjugate the passengers with the aid of knives, box cutters, a gun, and a device with wires that was said to be a bomb. Let's suppose the chance of a successful connection via cell phones is 1 in 20. What is the probability of at least 20 successful connections from 100 attempts? The undiluted probability is 1 in 20 ^ 20 = 1 in 1.05 * 10 ^ 26. Since the improbability of each event is only 20 and nCr for n = 100, r = 20 is very high at over 10 ^ 20, the nCr approximation grossly exaggerates the improbability diluting effect of the 80 failed connections. Using the nCr approximation, then Pdiluted = (1 in 1.05 * 10 ^ 26) * (nCr for n = 100, r = 20) = (1 in 1.05 * 10 ^ 26) * 5.36 * 10 ^ 20 = 1 in 195,630 A calculation using the exact algorithm shows the nCr approximation has over-corrected in this case by a factor of nearly 50. The actual probability, after assuming 80 unsuccessful attempts to call, is only around 1 in 9,503,036. Alternatively, suppose there were only 13 successful cell phone calls from 100 attempts, but the probability of a successful connection is only 1 in 100. The uncorrected probability is 1 in 100 ^ 13 = 1 in 10 ^ 26. Multiplying this by nCr for n = 100, r = 13, the adjusted probability becomes 1 in 1.41 * 10 ^ 10. An exact calculation shows this to be within an order of magnitude, since the actual probability is 1 in 3.16 * 10 ^ 10 after assuming 87 failed connections. The serendipity of incriminating evidence immediately after 9/11. Luckily for the authorities, a driver at Boston's Logan airport just happened to have an argument over a parking space with several Arabic-looking men. Improbable cell phone calls had provided the first evidence of an Arabic connection. The non-Arabic driver heard the news and contacted the authorities, who subsequently found an Arabic flight manual inside the abandoned white Mitsubishi. It was quickly discovered that the offending vehicle had been rented by Mohamed Atta. Atta was the only passenger (of 81 aboard Flight 11) whose luggage didn't make the flight. One bag contained a leather-bound Koran, a navy suit and a bottle of cologne. The other contained a videotape and flight manual for a Boeing 757, an Arab-English dictionary, and a manual slide-rule device known as a "flight computer".
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