Tuesday, July 8, 2014

Probability and Frequency

In the Theory of Probabilities we deal with a concept of a random experiment which results in occurrence or not occurrence of certain events.

For the purposes of this introduction into Theory of Probabilities we assume that all these experiments are independent in a sense that the result of one experiment does not affect the result of any other.

We also assume that these experiments are repeatable, so we can repeat the same experiment under similar conditions with, possibly, different results.

For example, consider a random experiment of shuffling and dealing a standard deck of 52 cards among four players, 13 cards each. An event we are interested in is to have four aces distributed to four players in such a way that each player has one ace. This event might occur or not occur as a result of our experiment.

Another important concept is elementary events from the combinations of which any other event can be constructed. In the example above each individual distribution of 52 cards among four players, 13 cards each, constitutes such an elementary event and we can construct the event we are interested in as a combination of particular distributions of certain cards to certain players. As we calculated while discussing Combinatorics, the number of these elementary events (that is, particular distributions of 52 cards among 4 players, 13 cards each) is 52!/(13!)4.

What's important about this and all other experiments we will be discussing is our ability to repeat this exact experiment any number of times, so we can count how many times the event in question occurred and how many times it did not occur, calculating its frequency of occurrence as a ratio between the number of experiments the event occurred to a total number of experiments (sometimes expressed as a percentage).

Presumably, if we repeat our experiment more and more times, the frequency of occurrence of a certain event will be more and more close to a certain limit value that can be called the probability of this event. This is not a precise mathematical definition of probability, but a rather philosophical explanation. It obviously depends on existing of this limit value, which we have no means to prove using just the above explanatory approach, but rather assume it. Precise mathematical foundations of this approach provide a solid base for this assumption.

A very important characteristic of our experiments is statistical similarity or inner symmetry of elementary events as their results. These elementary events have equal chances to occur and, by combining them in some way, we can construct other events we are interested in.

For example, since there are C(49,6) statistically equivalent results of choosing 6 winning numbers out of 49 in lottery, we can safely assume that the frequency of occurrence of one or any other particular set of 6 winning numbers will tend to 1/C(49,6) as the number of experiments (randomly picking 6 numbers out of 49) increases to infinity. Therefore, the probability of any one particular set of 6 winning numbers equals to 1/C(49,6).

Notice that any event we are interested in (like each player has exactly one ace or the dice has an even number on top, or you guessed two out of six winning lottery numbers) is a combination of certain number of equally probable elementary events symmetrical in the course of the experiment.

As you see, two characteristics are very important to understand the probability from the point of view of frequency - repeatability of the experiments under the same conditions and inner symmetry of the elementary events occurring as a result of the experiment and from which we can "construct" any event we are interested in.

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