2. [A Real-World Scientic Application of a Binomial distribution] Dowsing rods are a nonsensical folk belief, which claimed that unseen underground water sources could be found by holding a twig. The water source supposedly would be magically 'sensed' by the twig, which was supposed to swivel in the hand when water was nearby. The notion is not true (and there's no physical reason why it should be). Nevertheless, untrue beliefs are often repackaged by the unscrupulous and sold to the desperate. A particularly shocking example involved a fraudulent device marketed to the military during the Iraq war which was claimed to 'sense' hidden roadside = 7 h bombs by some unknown means. The device, named the MOLE, was produced by Global Technical Ltd., UK. It had an antenna which was free to swivel, and it was claimed that the antenna would Swivel when explosives were nearby. There were elementary electronic components inside the device, but they were The MOLE device not connected in any meanrngful way, there was no battery, and there is no with swiveling aerial known technology that can detect explosives at such distances. (Photo from Saudi\" Natiml Labs publication, reference below) So the device was essentially nothing more than a modern dowsing rod, a scam device being marketed to soldiers who undoubtedly would have needlessly died if they had trusted the device. Many similar devices were marketed around the same time, and the Iraqi government actually spent 85 million dollars on one such worthless device. [You are encouraged to learn about this at the following link: http:/fnewsbbcco .uk/Z/hi/programmesewsnightf 8471 187 .stm] Luckily, the US. Sandia National Labs were skeptical about the MOLE device, so they conducted a test. [\"Double Blind Field Evaluation of the MOLE Programmable Detection System,\" Dale Murray, Sandia National Laboratories; a copy is posted on Canvas for interested students] In the test, 20 grams of C4 explosive were placed in one of four possible boxes at the test facility. A representative from the Global Technical Ltd company, who claimed that the device really worked, would then use the detector to attempt to nd which box contained the hidden explosive. The test was double-blind. [exercise 2 continues on next page] 2 (a) [3pts] If the operator of the device during the test was essentially just guessing completely at random regarding which of the four boxes contained the explosive, what is his probability p of getting the right box? The test was conducted a total of 20 times (with the explosive placed in a different box each time). Consider the number of times X the operator of the device could have correctly found the explosive, out of the 20 trials. Then X is a Binomial random variable, with constants n and p. 2 (b) [Spts] If the device didn't really work, and if for each of the 20 'runs' of the study the operator of the device was therefore essentially just guessing completely at random regarding which of the four boxes contained the explosive, state It and p for this scenario (be sure to say which is which). 2 (c) [lpts] Use the Binomial Formula (hint, see pp. 45 of lecture 14, or part II of lab 5) to compute P(X=6), the likelihood of guessing right in exactly 6 of the 20 trials. For the purpose of the exercise, show the formulation with all the relevant values pluggedin, then give the answer. 2 (d) [4pts] Does the probability in part c represent something that would be relatively surprising to happen just by random, or is it so relatively large that it represents something that is not surprising to happen just by random? [If you're unsure, look back at what was said on page 2 of lecture 9.]