BENFORD'S LAW - Saurabh Gopal Agrawal 2013038

Saurabh Gopal Agrawal

2013038

Group - 6

BENFORD'S LAW:

                               

Benford’s Law is one of those mathematical laws that seems to defy common sense but works for most naturally occurring number sets.

It says that in most groups of naturally occurring numbers, the leading digit 1 will occur more than 2 as a leading digit and so on down to numbers starting with 9 occurring least often.

BENFORD’S LAW IN EXCEL:

Firstly, create a column of leading digits only using the LEFT() function. Despite what Excel documentation sometimes says, LEFT() works with numbers (not just text) and will ignore any currency symbol if defined in the cell formatting. For Benford’s Law use LEFT(<value>,1)

Then use COUNTIF to count the instances of each leading digit from 1 to 9 e.g. COUNTIF(<leading digit>,”1”) – remember that LEFT() returns a string/text value so the COUNTIF comparison is “1” not the digit 1 .

For a more exact set of comparison values use the formula =LOG10(1/<leading digit>+1).

 EXAMPLE WORKSHEET:

The leading digit values are shown in a separate column.

USE WITH CARE:

In the real world, Benford’s Law is often applied to check if data has been tampered with or outright made up. If someone has faked data or tinkered with the numbers that will affect the Benford’s Law distribution. This makes it a useful tool for auditors or others checking for fraudulent data.

But Benford’s Law needs to be used with care because not all data sets are distributed evenly or widely enough.

An example that would NOT work with Benford’s Law is a list of petty cash receipts, because the petty cash limit might be say $40 so most of the amounts will have leading digits between 1 and 3 only and probably many just under the $40 limit. Similarly a list of large check approvals, because of the arbitrary definition of ‘large’ in any organization. However if you had a list of all outgoings from small to large, Benford’s Law might apply.

A series of adult human heights or weights also don’t obey Benford’s Law because most people are within a narrow range of heights or weights (i.e. you won’t have adults weighing 10lb or 20kg). Telephone numbers won’t work because there are arbitrary prefixes or blocks of numbers issues. On the other hand a large list of street numbers from an address list probably will obey Benford’s Law.

In short, Benford’s Law is a useful tool for checking data, but it needs to be used with care and understanding of the data source. Large scale numbers without arbitrary limits work best. A history of Benford’s Law is littered with people who falsely claim fraud based on a mistaken understanding of the data source.

 Source: http://news.office-watch.com/t/n.aspx?a=1748

YESHA SHAH
2013039
GROUP 6

Today, we played a card game in group. The game we played is Kruskal’s Count. 

Kruskal’s Count

This trick may be perform to one individual or to a whole audience, and involves the spectators counting through a pack of cards until they reach a final chosen card. Yet, despite this seemingly random choice of cards, the magician is still able to predict the spectator’s chosen card. The trick is known as ‘Kruskal’s Count’ and was invented by the American mathematician and physicist, Martin Kruskal and described by Martin Gardner. Although this trick will not work every time, we will show that the probability of success is around 85%.

The Trick
A spectator is invited to shuffle a pack of cards as many times as they like. The spectator is then asked to secretly pick a number between 1 and 10 and to count along as cards from the deck are displayed. The magician may choose to display the cards one at a time, or he may choose to display all 52 cards together. The magician explains that the card in the position of the spectator’s secret number becomes the spectator’s first chosen card. The spectator is then told to use the value of that chosen card as his new number, and to repeat the process until the magician runs out of cards. Here, aces are worth 1; Jack, Queen, King are worth 5; and all other cards take their face value.
Yet, despite this seemingly random path through a shuffled pack of cards, the magician is able to predict
the spectator’s last chosen card. Watch and interact with a video of the trick being performed here.

The Secret
How is this done?

Well, unknown to the spectator, the magician also picks an initial number between 1 and 10, and proceeds to go through the same process. And although the magician may not have picked the same number as the spectator, there is a high probability they will land on the same final card. This is because, even though the magician and the spectator begin on different paths, there will come a point, simply by coincidence, when the two players land on the same card. And from that point on the two paths will become synchronized, meaning both players end on the same final card.

If we assume the initial numbers are equally likely to be chosen, then the probability
of success is 84%. And we can increase that chance slightly, to 85%, if the magician chooses 1 as his initial
number.

Furthermore, we will now show that, if N is the number of cards, and x is the mean average card value,
then the probability of success may be approximated with the simple formula

This is an illustration of the game, where two different situations are showed. The first situation is when the player initially starts with number 1 in his mind and then plays the game, which is denoted by yellow dots. Similarly in the second situation the player starts the game with number 7 in his mind and then plays the game, which is denoted by blue dots. The last card which is left is 8 of hearts, in both the situations, which is 85% probability. 

Source: http://www.singingbanana.com