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# Probability and Permutations

## Determine the probabilities of events when arrangement order matters

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Error Control Codes

Credit: Laura Guerin
Source: CK-12 Foundation

Have you ever experienced a dropped cellular call? How about distorted audio? All our electronic information is transformed into bits and bytes. But did you know that a large portion of these bits get lost and distorted when they’re sent between devices? If it wasn’t for some strong error detection and recovery codes, we wouldn’t have the wireless systems that we have today.

#### Why It Matters

The wireless interface distorts the signals that carry all the bits that make up your conversations, messages, and everything else that you upload or download. If bits have errors, your information can be misinterpreted. These bits therefore are encoded with error protection capabilities. For example, let’s assume that on average, 1 out of every 10 bits that are sent is erroneous (the bit ‘1’ turns into ‘0’ or the ‘0’ into a ‘1’). The probability that a certain bit has an error is therefore 110=0.1\begin{align*}\frac{1}{10}=0.1\end{align*}. This is called the bit error rate (BER). How do you improve the reliability of your bits? How do you correct, or at least detect, an error?

Credit: Brett Jordan
Source: http://www.flickr.com/photos/x1brett/6665955101/

Try this. For every bit that you need to send, send it three times instead of once. So when you need to send ‘1’, you actually send ‘111’. If the receiver at the other end receives a ‘111’, that′s great. But if there is a 1-bit error, and the receiver receives ‘110’ or ‘101’ or ‘011’, it can still assume that ‘1’ was the correct bit. The receiver will wrongly interpret the message as a ‘0’ only if it receives any of these: {000, 001, 010, 100}. The probability of an erroneous interpretation (the BER) becomes 0.028, which is lower than 0.1. This is an example of a repetition code.

The error control codes used in reality are much more complex than this. Nevertheless, another simple method is to use parity bits.

See for yourself: http://en.wikipedia.org/wiki/Parity_bit

#### Explore More

Show how the coded BER is 0.028 for the repetition code described above.

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