Randomness

Pronunciation: /ˈɹæn'dəm/ Explain

Random means without pattern and non-repeating. A random number is a number that is selected from a list of numbers where the list does not have a pattern and does not repeat. A random event is an event that is not part of a pattern of events. In some contexts, randomness is called 'noise'. Randomness is important in probability, statistics, quantum mechanics, and other fields in science: Probability: Lists of random numbers are used to simulate real-life events. Statistics: Random numbers are used to help randomly select members of a population into a random sample. Quantum mechanics: Many physicist believe that at a sub-atomic scale, events are random. Encryption: Measures of randomness are used to measure the quality of an encryption scheme.

Random numbers

A list of numbers is considered random if no pattern can be found and if the numbers satisfy no algorithm. Realistically speaking, the total number of patterns is infinite and the total number of algorithms is infinite, so lists of random numbers are compared to a set of well-defined number patterns, and evaluated against a set of well-defined algorithms. For more on testing the randomness of a sequence of numbers, see Testing Randomness in this article.

Random number generation

Since random numbers are used extensively in probability, statistics and various sciences, the generation of random numbers is an important topic. There are two types of random number generators: pseudo random number generators and true random number generators. A pseudo random number generator generates seemingly random lists of numbers from non-random events. The rand() and random() functions in most computer languages are pseudo random number generators. They use various mathematical techniques to generate numbers that have a high degree of randomness. However, research has shown that these lists are not truly random, but are pseudo random.

A true random number generator generates lists of numbers that are truly random. Two of the most common types of true random number generators use the time difference between natural events. One uses the elapsed time between radioactive decay of atoms in a radioactive sample. The other uses the elapsed time between electrical atmospheric events.

While, on the average, the total number of such events in any relatively large time frame is predictable, the exact amount of time between events is random. One free online service, Random.org, uses a very sensitive device that detects atmospheric events to generate random numbers.

Discovery 1 - Your own pseudo random number generator

In this discovery, you will learn how to generate lists of random numbers. In personal computers, random numbers are often generated by counting the milliseconds or clock cycles between keystrokes. While people tend to push keys at regular intervals, the exact timing between keystrokes has a high degree of randomness.

This discovery requires a digital watch that displays seconds, a piece of paper, a pencil and something to write on such as a clipboard or a book. Start by picking a common sound at the start of words, such as the 'g' sound in 'great'. Go to a place where you can listen to conversations. Every time you hear a word that starts with the selected sound, write the seconds down on the piece of paper. For example, if someone says 'green' and the time is 11:36:57 write down 57. When you have written down a lot of numbers (at least 32, better yet 100), take a look at your list. Do you see any pattern to the numbers? You can use the randomness tests in Discovery 2 to test your list for randomness.

Testing Randomness

Because of the importance of randomness in the sciences, there are many and extensive randomness tests have been developed to check any particular list of numbers for randomness. Most of these tests require a lot of computer power and are way beyond the scope of this encyclopedia. However, there are a few easy tests that can be performed on small lists with pencil and paper.

One such test is the oddness or evenness of the numbers in the list. If a list is truly random, about one-half of the numbers will be even and one-half odd. If the count of odd numbers and even numbers varies greatly from half and half, the randomness of the sequence is suspect. One the other hand, the probability of the count of odd and even numbers being exactly half and half is very low. Having the count of even and odd numbers be exactly half and half is also suspect.

Another test involves checking whether two consecutive numbers are both even or both odd, or if the oddness of the numbers change from one number to the next. Given two consecutive numbers from a list of random numbers, the oddness of the numbers should change about half the time and remain the same about half the time. Again, if the count of changes versus non-changes is not close to half and half, or is too close to half and half, the randomness of the list is suspect.

Discovery 2 - Testing randomness of a random number generator

Use the true random number generator on this page to complete this discovery. You will need a piece of paper and a pencil. Alternatively, you can print out the worksheet at http://www.lifeisastoryproblem.com/worksheets/randomness.pdf. You will be using the true random number generator on this page to complete the discovery. You will be writing down 100 random numbers, then use the oddness and evenness of the numbers to verify that the list is a random list. In the context of probability, 100 numbers is a small list. There is a small but significant chance that your list will not seem random.

1. Divide your paper into three columns. Label the columns 'Number', 'Odd/Even' and 'Changed'. To fit in all 100 numbers, you will need two or three sets of columns.
2. In the true random number generator, type 1 for Min and 100 for Max. This will cause the random number generator to create numbers between 1 and 100.
3. Click the Generate button. Write down the number that appears in the column labeled 'Number'.
4. Repeat the previous step until you have written down 100 numbers.
5. For each of the numbers you have written down, write 'O' for odd or 'E' for even in the column labeled 'Odd/Even'.
6. In the column labeled 'Changed', for each of the numbers after the first, write 'Y' if the oddness of the previous number is different from the oddness of the current number. Write 'N' if the oddness of the previous number is the same as the oddness of the current number.
7. Write down the number of even numbers and the number of odd numbers. Calculate and write down the percentages.
8. Write down the number of times the evenness changed and the number of times it remained the same. Calculate and write down the percentages.
9. The percentages should all be close to 50%. With only 100 numbers to compare, it is likely that the percentage is between 45% and 55%.

Random numbers in probability

In probability, random numbers are used to simulate real life events. For example, to simulate the birth of a boy or a girl, one might use a random number generate where an odd number simulates the birth of a boy and an even number simulates the birth of a girl. For more information see Simulation.

Random numbers and statistics

One of the most important steps in completing a survey is selecting a random sample. Random numbers are often used to select members of a population to survey or to select a region to survey. For example, when doing exit polls during elections, the actual poling places surveyed sampled may be selected randomly from all the poling places available. Then, all of the people voting at that poling place are included in the sample.

Randomness and encryption

When coming up with an encryption scheme, extensive testing is used to make sure that the encryption is hard to break. One test is the randomness of the encrypted data. If a pattern can be detected in the encrypted data, it is possible that the pattern can be used to break the encryption. Additionally, if the encryption key is changed there must be no pattern to the changes in the encrypted data.

One test performed when validating encrypt schemes involves changing one bit (a bit is a 1 or a 0) in the encryption key. The scheme passes this particular test if about one-half of the bits in the encrypted data change. There are many other test that are also performed.

References

1. McAdams, David E.. All Math Words Dictionary, random. 2nd Classroom edition 20150108-4799968. pg 150. Life is a Story Problem LLC. January 8, 2015. Buy the book

Cite this article as:

McAdams, David E. Randomness. 3/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/r/randomness.html.

Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
12/4/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)