Startup, Entrepreneurship and Certification. You will also find the trending articles

2012
Twitter Facebook



USB Flash drive Serial Numbers - "UNIQUE"?

Formatted USB flash drives (a.k.a. thumb drives etc) have Volume Serial numbers generated when the new filesystem gets created. The algorithm depends on a file system and OS. Volume Serial number can easily be changed via hex editor at locations:

FAT 12/16 - 4 bytes at offset 0x027
FAT 32      - 4 bytes at offset 0x043
NTFS         - 8 bytes at offset 0x48

or by using a myriad of free tools that can be found on the Internet. Volume Serial numbers are important from the forensic investigations stand point and there have been plenty of good material written on this topic. The most prominent in my view are written by Craig Wilson, Rob Lee and Harlan Carvey.

Unlike Device Serial Numbers, Volume ID's get captured by all forensic imaging tools.Device Serial Numbers however have been considered by computer forensic practitioners as more reliable and "Unique" artefacts. In Windows there are several places when Device Serial Numbers get recorded/logged. USBStor  registry key and Windows log files: Setupapi.log on Windows XP or Setupapi.dev.log on Vista and above are the most obvious one. It is also a well known fact that when a USB flash drive has no serial number, the system assigns to the device its own number with an ampersand symbol as the second character of this serial number.


The question is, how "UNIQUE" these Device Serial numbers are?
Well, as it turns out, these numbers are not necessary unique. There could be several reasons for this.

1. There is a tool that gamers are using to spoof device serial numbers called PB DownForce. It is capable of temporary changing the device serial number. The serial number can be changed to a random or predefine serial number.

This wont fool (see picture below) tools like USBDeview, but the software that rely on Operating System to obtain the serial number will fall for it.

2. USB drives serial numbers are meant to be at least 12 valid characters, represented as a UNICODE string.  "The last 12 digits of the serial number shall be unique to each USB idVendor and idProduct pair" according Universal Serial Bus Mass Storage Class paper.

   Valid Serial Number Characters

        Numeric                        ASCII
0030h  through 0039h      "0" through "9"
0041h through 0046h       "A" through "F

These requirements have not been adopted as the mandatory standard and a lot of manufacturers use shorter and in many cases identical numbers on their cheaper drives.

3. Big labels do use "unique" serial numbers, especially on their upper class, higher capacity USB devices. Still some reuse serial numbers every 6th million times as in case with one of the popular USB storage manufacturer I had to deal with.

4. Devices can be FAKE. On eBay there are plenty of fake 'false capacity usb flash drives', including brand name counterfeits such as 16GB Kingston, 32GB Sandisk etc. Serial numbers on these devices can be ether, all identical or generated at random.

5. User can change the device serial number accidentally or on purpose.  There are many tools, mostly used to fix faulty USB flash drives, capable of changing the device serial number. FixFakeFlash Inspectortech website is a good place to learn more about fake USB devices and tools capable of changing many parameters on the USB device including the serial number, ability to create, encrypt, hide or write protect certain area on the device.




The above-mentioned tools are designed to work with different USB flash drive controllers and you of course must have the right one to be able to reprogram the device.
Names of memory controllers can be coded in the original (Factory set) Serial Number. For example some Kingston's devices in13th position of the serial number have a letter A, B, E, C or F:

Kingston DataTraveler 200 USB Device SN: 001A92053B6ABB4131340023

A        - SkyMedi
B or E - Phison
C or F - SSS
To my knowledge similar tools are available for the memory controllers listed below:
  • Alcor
  • Ameco (MXTronics)
  • Chipsbank
  • iCreate
  • ITE tech
  • Netac
  • OTI
  • Phison
  • Prolific
  • RAMOS
  • Skymedi
  • SMI (Silicon Motion)
  • SSS (Solid State System)
In addition to USBDeview there is another excellent tool called ChipGenius (by Chinese Developers at mydigit.cn) that provides a lot of useful information about a USB Device. The tool can be used to check pretty much all types of USB devices including external hard drives and MP3 players, detect fakes and view the device controller vendor.




It displays chip model, manufacturer, revision number, VID/PID, interface speed, protocol, serial number and media type information.

Finally, unlike Volume Serial Numbers most forensic imaging tools don't capture a Device Serial number. The only exception to this rule I know are Tableau imagers. Bothhardware (TD1 & TD2 duplicators) and Software (TIM a.k.a. Tableau's High Performance Software Imager) include the Device Serial Number in the acquisition log automatically (but not in the image itself).

Speaking about Tableau devices, a new generation of TD duplicator TD2 is looking really sexy. TD1 has been used by my team quite extensively. The new version "can optionally include USB, SCSI and SAS suspect drive" and what is even more exciting is the ability to image 1:2 or how Tableau called it "Twinning" support. According to the specifications it also supports EnCase v7 .ex01 (AES encrypted) format. I definitely going to order one of these very shortly.
November 19, 2012
Twitter Facebook




Despite what you may have heard, the Web is not the Wild West. This isn’t ‘Nam, as The Big Lebowski‘s Walter Sobchak so wisely proclaimed. There are rules. And to be a responsible Web citizen, it is important for us to understand what these rules are. To get a better sense of the legal landscape, we spoke with Trevor Timm, an activist with the Electronic Frontier Foundation, who provided his insights on the handful of laws every Internet users needs to know.

1. Digital Millennium Copyright Act

DMCA-scroll

What it is: Enacted in 1998 by President Bill Clinton, the Digital Millennium Copyright Act (or DMCA) is the primary law governing the use of copyrighted content in the U.S. It establishes a system for removing copyrighted works that are being used improperly, and dictates who’s responsible when this happens (as it so often does).
Why it’s good: “By far, the best part of DMCA is its ‘safe harbor’ provisions, which basically states that websites aren’t liable for for the copyright infringement of their users, as long as they meet certain requirements,” says Timm. “That part of the DMCA is what has allowed companies like Google and Facebook to thrive over the last decade.”
Why it’s bad: The takedown system of DMCA is good in that it requires rightsholders to go after specific pieces of copyrighted content, says Timm. “But the problem is, it can be abused — it’s being abused more than it ever has in history, right now.” This is because websites must take down content for a certain period of time, regardless of the validity of the claim, in order to securely keep their safe harbor protections. As a result, websites “always err on the side of the copyright holder,” says Timm, which can lead to legitimate free speech being censored for as much as two weeks without recourse — plenty of time to keep bad information out of the news cycle.

Law in action: Your YouTube video is taken offline because it includes a Justin Bieber track playing in the background.
Bottom line: DMCA has its faults — but it’s much better than some of the alternatives that have come before Congress.

2. Electronic Communications Privacy Act of 1986

What it is: Enacted in 1986, the Electronic Communications Privacy Act (ECPA) extended the prohibition of government wiretap laws from phone lines to computers. The law governs everything from email to instant messages to cloud storage files.
Why it’s good: ECPA is a good example of the government doing the right thing to protect its citizens from unreasonable searches and seizures.
Why it’s bad: “It was written before the World Wide Web was invented,” says Timm, so it’s now woefully out of date. For example, ECPA mandates that an online communication or file that’s held by a third-party (like Google, Facebook, or Dropbox) for more than 180 days is “abandoned,” since, at the time it was written, Web-based email was still a novelty. But because of this provision, your six-month-old communications and files may be accessed by the courts with only a subpoena rather than a probable-cause warrant.

Law in action: Law enforcement agents gain access to the last three years of your Gmail communications as a routine part of an investigation.
Bottom line: A number of Congressmen are working to update ECPA to eliminate the 180-day abandonment provision, which would in turn require police and prosecutors to obtain a warrant before snooping your chat logs.


3. Patriot Act

What it is: A monster piece of legislation, the Patriot Act is the paramount anti-terrorism legislation in the U.S. — and easily one of the most controversial laws on the books. For Internet users, three mechanisms of the Patriot Act are of foremost importance: national security letters, pen register, and trap-and-trace orders — legal mechanisms we often refer to as warrantless wiretaps.
National security letters are used by organizations like the FBI to obtain information about your communications, such as who you’re contacting and how often, without the need for a warrant. Both pen registers and trap-and-trace orders are also used to gather this information — sender and recipient names, phone numbers, email address — in real-time. No probable cause warrant is needed, since neither access the contents of messages, only the messages’ surrounding information.
Why it’s good: Champions of the Patriot Act argue that, prior to its passage, the wiretap laws in the U.S. were outdated, and that tools like pen registers and trap-and-trace allow law enforcement authorities to better protect U.S. national security in a more connected world, where suspects have a greater number of ways to communicate.
Why it’s bad: In general, the Patriot Act has greatly reduced the privacy and Fourth Amendment rights protections for Americans and the people with whom we communicate. National security letters, pen registers, and trap-and-trace lie at the heart of that erosion, since they lessen judicial oversight for what amounts to wiretapping. Documents obtained by the ACLU at the end of September show that use of pen registers and trap-and-trace has increased more than 600 percent since 2001, as of the end of 2011.
Law in action: The FBI demands to see all your phone cell phone records for the past five years from your wireless provider, no questions asked.
Bottom line: National security is an issue of indisputable importance. But in the case of the Patriot Act’s warrantless wiretapping mechanisms, there’s no denying that it has limited U.S. civil rights in a profound and, many would say, unconstitutional way.

4. FISA Amendments Act

FISA-AA-scroll
What it is: The FISA Amendments Act is an expansion of the warrantless wiretapping powers granted to President George W. Bush. First passed in 2008, and up for reauthorization by Congress this year, the FISA Amendments Act specifically targets foreign intelligence, and removes the requirement for the U.S. government to specify who and what is being targeted  The purpose of these warrantless wiretaps is simply “to acquire foreign intelligence information,” according to the law’s text.
Why it’s good: Like the Patriot Act, supporters of the FISA Amendments Act — which includes a majority of Members in the House as well as the Obama administration — say that the law is needed to protect the U.S. from external threats to our national security.
Why it’s bad: “It allows the government to get these secret court orders, kind of like general warrants, where they can get hundreds — or potentially millions of people — under a dragnet-type surveillance for up to one year,” explains Timm. “So they could target whole countries that have anything to do with our foreign intelligence information.” In other words, “if you’re talking about politics on the phone, they could potentially wiretap your communications.” Critics warnthat the FISA Amendments Act essentially gives the U.S. government unlimited spying powers, on both foreign nations and U.S. citizens.
FISA Amendments Act champions say that concerns about the liberties of those who live outside the U.S. are invalid because foreign nations do not enjoy protections under the U.S. Constitution.
Law in action: The National Security Agency has had to build the largest data center in the U.S. to handle the massive influx of communications gathered under the FISA Amendments Act and other wiretapping mechanisms.
Bottom line: As with the Patriot Act, the FISA Amendments Act does appear to help protect U.S. national security. The problem is that it does so at the expense of individual liberties of countless people in the U.S. and around the world.

5. Communications Decency Act (Section 230)

CDA-230-scroll
What it is: Passed in 1996, the Communications Decency Act (CDA) as a whole was a wretched piece of legislation. And much of it has since been struck down by the Supreme Court due to its restrictions to free speech, thanks to efforts by the EFF and the ACLU. However, one important part, known as Section 230, remains. This provision removes the liability of “interactive websites” — think YouTube, Reddit, any blog, social network, or other site with commenting — for things said by Web users. In other words, if you write on Facebook that I’m a car thief when I’m not, Section 230 says that I can’t sue Facebook for your lie — I can only sue you.
Why it’s good: Like DMCA “safe harbor,” Section 230 has essentially allowed many of the most popular websites and online services to exist without being sued into oblivion. “Basically, it offers a legal shield to all sorts of bloggers, and is what has allow Google, and Facebook, and Craigslist, and all these sites to thrive,” says Timm. After all, “Facebook’s legal costs would be in the trillions of dollars if they had to worry about what 900 million people were saying.”
Why it’s bad: Critics of Section 230 argue that it is too protective, and leaves little recourse for anyone who is genuinely harmed by the comments of other Web users, especially in cases of anonymous commenting.
Law in action: The comments section of any website (especially YouTube).
Bottom line: Without CDA’s Section 230, the Web as we know it would likely not exist due to an avalanche of libel lawsuits.





October 04, 2012
Twitter Facebook


Instructions



    • 1
      For all problems of this type, there are some important points that apply, which are expanded upon in the article linked in the Reference section. First of all, the problem will probably refer to a "fair" die. This just means that no tricks are involved. It's not weighted.
      We also assume a normal situation. Don't question whether or not the die could come to rest on a corner or up against the wall, or anything like that. It will land on one of the six sides fairly. The problems typically refer to a standard six-sided die, unless you are told otherwise. Note that some textbooks may refer to a die as a "standard number cube," so that the publisher will not be accused of promoting gambling among children.
    • 2
      Here is a typical question: "A fair die lands on 1 five times in a row. What are the chances that it will land on 1 on the next roll?" The answer to the question is 1/6. That is it. Any other answer is wrong. Each roll of a die is totally independent. See the article in the Resource section for expanded information on that important concept.
    • 3
      Another typical question is, "What are the chances that a fair die will land on 7?" The answer is 0%. Since there is no 7 on a standard die, the outcome cannot occur. Similarly, if you are asked what are the chances that the die will land on 1, 2, 3, 4, 5, or 6, the answer is 100%. There are 6 possible ways for success out of 6 ways total. From that standpoint, we could write our answer as 6/6, which simplifies to 1.
    • 4
      Note that all probabilities are a number between 0 and 1, which we often convert to a percent by converting the fraction to a decimal, and then moving the decimal two places to the right. For example, 0 becomes 0%, 1/2 becomes 50%, 1/6 becomes 16.67% (rounded), and 1 becomes 100%.
    • 5
      Another possible question is: "What are the chances that a die will land on 2 or 3 when rolled?" We are dealing with an "or" condition, which means we must add the possible outcomes together (since either is OK). In this case there are two possible means of success, out of 6 total. Our answer is 2/6. Interestingly, in probability we would usually leave the answer just like that, rather than reduce it to 1/3 which would de-emphasize the denominator of 6 total possible outcomes.
    • 6



      Another possible question is: "A fair standard number cube is rolled twice. What are the chances that it will land on 2 the first time, and an odd number the second time?" In this case we have an "and" condition, separating two independent events. Each die roll knows nothing of the other. We use multiplication for this. For the first roll, the chances of success are 1/6. For the second roll it is 3/6, since there are three odd values on a die (1, 3, 5). Multiply 1/6 times 3/6 to get 3/36. In this case, we would probably reduce our answer to 1/12.
    • 7



      The final type of question discussed is: "A fair die is to be rolled 10 times. What are the chances that it will land on 3 every time? Express your answer using an exponent." The chances of a 3 on each roll is 1/6. We're dealing an "and" condition. Each roll has to be a 3. We must compute 1/6 times 1/6 times 1/6, repeated a total of 10 times. The simplest way of representing this is shown at left. It is (1/6) raised to the 10th power. The exponent is applied to both the numerator and the denominator. Since 1 to the power of 10 is just 1, we could also just write our answer as 1 divided by (6 to the 10th power).
    • 8
      It is interesting to note that the actual odds of the above happening are about one in a 60 million. While it is unlikely that any one particular person will experience this, if you were to ask every single American to conduct this experiment honestly and accurately, there is a good chance that a few people would report success.
    • 9
      Students should make sure that they are comfortable working with the basic probability concepts discussed in this article since they come up quite frequently.

July 14, 2012
Twitter Facebook

How Throwing Dice theory work

Single dice

Click here for how to write a probability.
Throwing dice is more complicated than tossing coins, as there are more than 2 values. If you throw a single dice, then it can fall six ways, each of which is equally likely if the dice is true. So the probability of getting one particular value is 1/6. If you want either of two values it is 2/6 or 1/3, and so on.

Sum of two dice

It gets more interesting when you have two dice. One thing that you can do is work out what the total of the dice is. The dice experiment allows you to simulate throwing pairs of dice and see what the result is. This is a good introduction to probability, since you can see which combinations are more likely. But the real world, or even a simulated real world, never matches completely with calculated probability. So how do we calculate it? The first thing is to work out what the range is. You can't have a total less than 2 (both dice being 1) and you can't have a total more than 12 (both dice being 6). The easiest way to see what the probabilities is to write out the possible totals. There are 36 of them in all (6 x 6).
Total on dicePairs of diceProbability
21+11/36 = 3%
31+2, 2+12/36 = 6%
41+3, 2+2, 3+13/36 = 8%
51+4, 2+3, 3+2, 4+14/36 = 11%
61+5, 2+4, 3+3, 4+2, 5+15/36 = 14%
71+6, 2+5, 3+4, 4+3, 5+2, 6+16/36 = 17%
82+6, 3+5, 4+4, 5+3, 6+25/36 = 14%
93+6, 4+5, 5+4, 6+34/36 = 11%
104+6, 5+5, 6+43/36 = 8%
115+6, 6+52/36 = 6%
126+61/36 = 3%
The percentages above are rounded. Compare these with the dice experiment. It won't match exactly, although the more throws you do, the closer it will get.

Other conditions with two dice

There are plenty of other conditions that you can have with two dice. Often, the easiest way to get the probability of a condition is to list all possible throws, and count the ones that fit the condition. The best way to make sure you have remembered all throws is to list them in a 6 x 6 table, like below.
Imagine two dice, red and green. This table shows all possible throws of these dice, with red numbers for the red dice, and green for the green dice. The background is black if the condition is satisfied. You can change the numbers and the condition to see what happens. You find the probability of this condition by counting the black squares, and dividing by 36 (all possible throws).
First number:

 Either dice is 3
 Neither dice is 3
 Both dice are 3
 At least one dice isn't3
 The highest dice is3
 The lowest dice is 3		Second number: 

 Dice are3 or 5
 Dice are not3 and not 5
 Dice are3 and 5
1 11 21 31 41 51 6
2 12 22 32 42 52 6
3 13 23 33 43 53 6
4 14 24 34 44 54 6
5 15 25 35 45 55 6
6 16 26 36 46 56 6
Probability of this condition =11/36See below for an explanation
or click here

Either dice is a particular number


The probability of one dice being a particular number is 1/6. You would assume that it would be twice as likely that either of two dice being a particular number, or 1/3, but this would be wrong. If you select the first condition above, you will see why. While there are six conditions where one dice is a number, and six conditions where the other dice is that number, there is one condition in both. So the probability is (12-1)/36 or11/36.
There is a better way to calculate this. If you click on the second condition, then back on the first, you will find that they are opposites of each other. That means that their probabilities add up to one. The second condition is easier to calculate directly (see below), as 25/36. So this probability is (36-25)/36 = 11/36.

Neither dice is a particular number


The probability of one dice not being a particular number is 5/6. The probability of two dice not being a that number is 5/6 x 5/6 = 25/36. You can do this as they don't overlap as all (the correct term is that they are independent events).

Both dice are the same particular number


The probability of one dice being a particular number is 1/6. The probability of two dice being the same particular number is 1/6 x 1/6 =1/36. This is not the same as saying that both dice are the same number. There are six different possible numbers, so that would be 6/36or 1/6.

At least one dice isn't a particular number


This is the opposite of both dice being the same particular number, so the probabilities will add up to one. So the probability of at least one dice isn't a particular number = 1 - (1/36) = 35/36.

The highest dice is a particular number


This is a fun conditions, but a little strange to work out! Try experimenting with different numbers, and you will get different collections of throws fitting the conditions, and so different probabilities. To take an example, considering one dice, if the particular number is 5 then there are 5 throws which fit the condition. There are another 5 for the other dice, and there is one overlap, so the result is (5 + 5 - 1)/36 = 9/36 or 1/4. But this varies for other numbers.

The lowest dice is a particular number


The probabilities vary considering what the number is. But the probability of the lowest dice being a particular number is the same as the probability of the highest dice being six minus that particular number.

The dice have two particular numbers


Like the first condition, there are two ways to work this out. The first is doing it directly. Each dice has 6 throws for each number, but there are four overlaps. So the probability is (4 x 6 - 4)/36 = 20/36.
An easier way is to realise that this condition is the opposite of the next one, dice are not either number. That probability is 16/36 so this one is 1 - 16/36 = 20/36 or 5/9.

The dice are not either particular number


The probability of one dice not being either of two numbers is 4/6, so the probability of both dice fitting this condition is 4/6 x 4/6 = 16/36. We can do this as the conditions don't overlap, so they are independent events.

The dice are two particular numbers


There are only two ways that two dice can have two values (try the exercise above to see why), so the probability is 2/36 or 1/18.

More than two dice

As the number of dice increase, then the different conditions can become more complicated. You can work out some simply.
The easiest is the probability of a number of dice being a particular number. For n dice, this is 1/(6n). For example, the probability for throwing 5 dice and getting them all sixes is 1/(65) = 1/(6x6x6x6x6) =1/7776 = 0.000128 or 7775 to 1.
You can work out what is the chance of getting various numbers when throwing several dice (or one dice several times), but you do it in a back-to-front way, as above. For example, if you throw three dice, what is the chance of one of the dice getting less than five? This means that one of the dice (at least) will be a one, two, three or four. Now this is quite hard to work out directly. What we can do is work out the opposite. The chance of one dice not getting less than five means the chance of it getting a five or six, and this is 2/6, or 1/3. But we need all dice to get this. This is 1/3x1/3x1/3 or 1/27. This was the opposite of what we want, so we must take it away from one. 1 - 1/27 or 26/27 which is about 96%.

Die or dice

People occasionally point out to me that the singular of 'dice' is 'die'. Yes, I know that. However, a lot of people don't, and I prefer to use 'dice' since then my meaning is clear. It stops them wondering why I'm talking about dying! However, if you think that I should be accurate rather than comprehensible, I'm afraid that the authorities are not on your side. The Oxford English Dictionary says "The form dice (used as pl. and sing.) is of much more frequent occurrence in gaming and related senses than the singular die." I think that probability is related to gaming! It gives a quote: 1474 CAXTON Chesse 132 "He caste thre dyse and on eche dyse was a sise." (Translation - "He casts three dice and on each dice was a six.")
July 14, 2012
Twitter Facebook


Probabilities are written as numbers between zero and one. A probability of one means that the event is certain. If you toss a coin, it will come up a head or a tail. So there is a probability of one that either of these will happen. A probability of zero means that an event is impossible. If you toss a coin, you cannot get both a head and a tail at the same time, so this has zero probability. Anything that can happen but is not certain is written as a number less than one. It could be a decimal, a fraction, a percentage, or described as "one in a thousand", which is another way of writing a fraction. All these are ways of describing probabilities. The coins experiment and dice experiment pages use percentages. Much of the theory is easier in fractions.
There are a couple of important points. Firstly, probabilities do not tell you what is going to happen, they merely tell you what is likely to happen. It is unlikely that you will toss twenty coins and that they will all come up heads. But if enough people toss enough coins for long enough, then this may well happen. It will startle the person it happens to, but think of all the people it didn't happen to! Secondly, if you toss a coin nineteen times and it comes up heads each time, then it is not more likely that the next toss will be a tail. The odds stay the same, at 50%. The tosses are called 'independent events' which means that the coin can't remember what has happened to it. While twenty heads in a row is unlikely, once you have nineteen heads in a row, the unlikely event has already happened. The potential twentieth head has the same probability as the first head. Another way of looking at it is that any sequence of twenty tosses is unlikely as twenty heads in a row, even if it looks random. But you have to write down the sequence before you start tossing to see if you get it!

What are the possible outcomes?

When we toss a coin, there are two possibly outcomes. It can be a head or a tail, which are both equally likely. If we toss two coins, there can be two heads, two tails, or a head and a tail. It is tempting to say that there are three equally possible outcomes. But this would be wrong. You must think of the coins separately. It might be easier to imagine tossing one coin first and the other after (or even tossing the same coin twice, which has exactly the same effect). Or you could imagine two different values of coins, so they can be told apart. Now you can see that there are four possibilities: both heads, both tails, first coin a head and the second a tail, and first coin a tail and the second a head.
Select number of coins to see all possible outcomes:       (H is a head and T is a tail)


How many possible outcomes?

You can see that the number of possible outcomes gets bigger and bigger. Click on One more coin to see how the number of possible outcomes increases:
    With 1 coin, there are 2 possible outcomes
It is quite easy to find the different outcomes, since they are represented by the binary numbers with that amount of digits, with H representing the digit one and T representing zero. This also shows us how many outcomes there are, since there are 2n possible binary numbers with ndigits.

Working out probabilies by counting

Once you have listed all possible outcomes, then you can work out the probabilities quite easily. Say that you are going to toss three coins, and you want to work out the probability of only one head (and so two tails). The possible outcomes are:
TTT, TTH, THT, THH, HTT, HTH, HHT, HHH
All these outcomes are different, and they are all equally likely. There are8 of them. There are 3 tosses with only one head:
TTH, THT, HTT
So the probability is 3/8. You can convert this into a decimal 0.375 or a percentage 37.5%, which you can round to 38% if you wish. Or you can describe it as a three in eight chance. All these mean the same.
You can list the possible outcomes above for any dice up to 6 and count the tosses which match the probability that you want. Then divide it by the total number of throws. Or you can use this calculator, where the computer counts the number of tosses for you!
Select number of coins:     Select number of heads:      
 
Now you can work out the probabilities for various combinations for heads and coins, then you can experiment to see if reality matches the probability! You will find it won't necessarily match (although the more numbers of throw, the closer it will get), but the experiments should give figures fairly close to the probability.

Calculating probabilities

The counting method works, and is very good for getting the right answer with a small number of coins. However, for larger numbers, we need a more mathematical approach. We know that the probability will be a fraction, and we know that the denominator (the number underneath) is2n for n coins. The problem is working out the numerator (the number on top). We don't want to count all the cases where it happens. What we can do is start with one coin, then add a coin at a time, and see what difference it makes to the probability. This starts to build up a pattern.
The probabilities for throwing a single coin are obvious:
if coin
is a head		if coin
is a tail		total of
these
chance of 1 head101
chance of no heads011
Now we add a coin. It will either be a head or a tail. We consider the probabilities separately. If it's a head, then the result for both coins will be either 2 heads or 1 head. If it's a tail, then the result will be either 2 tails (and so no heads) or 1 tail (and so 1 head). We can write down these probabilities, and get the final probability by adding them.
if new coin
is a head		if new coin
is a tail		total of
these
chance of 2 heads101
chance of 1 head112
chance of no heads011
You have to be careful about adding probabilities. Sometimes you can't, and if you try, you end up with a probability of more than one, which is not allowed - one means certain, remember! However, here you can. This means that we are considering the case of the new coin being either a head or a tail. The new coin can't be both, so these cases are independent of each other, which means that we are allowed to add the probabilities.
We add a third coin. The logic is the same, except we have more cases to consider. But you can see that we're just copying the probabilities from the previous case.
if new coin
is a head		if new coin
is a tail		total of
these
chance of 3 heads101
chance of 2 heads213
chance of 1 head123
chance of no heads011
We add a fourth coin.
if new coin
is a head		if new coin
is a tail		total of
these
chance of 4 heads101
chance of 3 heads314
chance of 2 heads336
chance of 1 head134
chance of no heads011
We add a fifth coin.
if new coin
is a head		if new coin
is a tail		total of
these
chance of 5 heads101
chance of 4 heads415
chance of 3 heads6410
chance of 2 heads4610
chance of 1 head145
chance of no heads011

Pascal's triangle

The pattern that emerges from the above example is something called Pascal's triangle (after Blaise Pascal a French mathematician). You work this out like this. Every line is made by adding the two numbers in the line above. You assume that there are zeros at the start and end of each line, and you start with a one in the top row. Pascal's triangle crops up a lot in probability. In particular, take the nth row of Pascal's triangle, and the mth number in it, and you have the numerator (top bit) of the probability of finding m heads when tossing n coins.
1
11
121
1331
14641
15101051
1615201561

July 14, 2012

Mr.16x9 blog

Contact Form

Name

Email *

Message *

Theme images by Jason Morrow. Powered by Blogger.
Javascript DisablePlease Enable Javascript To See All Widget