MATH 1342 Quiz 6- Page 2 of ?? ur number system consists of the digits 0, 1, 2,
ID: 3370788 • Letter: M
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MATH 1342 Quiz 6- Page 2 of ?? ur number system consists of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Because t numbers such as 12 as 012, the first significant digit in any mumber ,4, 5, 6, 7, 8, 9. Consider entries of numbers in accounting records, tax so that each of 2. (18 points) uumber we do not must be 1 write must be 1, 2, 3, 4, 5, 6 returns, etc. We may think that each digit appears the nine digits has a probability of being the first digit (in other words, with equal frequency outcomes), but this is not true- In 1881, Simon Newcomb discovered that first digits for numbers in records do not occur with egual frequency. The After studying lots and lots of data, he assigned probabilities the table below physicist Frank Benford discovered the same result in 1938. of occurrence as shown in role in identifying fraudulent data on tax returns and accounting bo dThe probablity law is now kmown as Benford's Lau and plays a major 0.097 0.079 0 0. 0.051 0.016 Table 1: Benford's Law (a) (2 points) Verify that Benford's Law satisfies our second probability rule, namely P(S) 1 0.201 + o. 176+0,125 + 0.0+0.079+ 0.067 +0.058+ 0.05)+ (b) (4 points) Use Benford's Law to the determine the probability that a randómly selected entry in some accounting books has a first digit that is a 1 or a 2. (c) (4 points) Compute the probability that a randomly selected entry in some ac- counting books has a first digit that i? 1 or 2 under the assumption that the digits are equally-likely. Rys|):u 1,l=0 2 How many more times is this event more likely under Benford's Law? 0.222 (d) (4 points) Use Benford's Law to determine the probability that a randomly selected eatry in some accounting books has a first digit that is at least 6, i.e. P(x 26). (e) (4 points) Use the result of part d. and the Law of the Complement (Rule 4 in our list of probability rules) to find the probability that a randomly selected entry in some accounting books is less than 6, i.e., P(XExplanation / Answer
b) Probability of first digit is either 1 or 2
P(X=1 or 2) = p(1)+ p(2)
= 0.301+0.176 = 0.477
c) b/0.222 = 0.477/0.222 = 2.149
d) p(X>=6) = p(X=7)+ p(X=8)+ p(X=9)
= 0.058+0.051+0.046
= 0.155
e) p(X<6) = 1-p(X>=6)
= 1-0.155
= 0.845
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