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Translations of Encyclopedia about Mathematics

 

Chance

Chance is understood as a randomly occurring set of events which is independent of our control. One of the most modern fields of mathematics is combinatory which, among others, is concerned with the numerical expression of chance.

Tools developed in the 16th and 17th centuries when studying games and which depended on chance gained importance only until the 21st century. There exist events which we can count on for certain based on the occurrence of a series of constant information (factors). For example, if we flip a coin, we can be sure that it will fall onto the ground (or in our hand). However, there are also events which are not foreseeable. For example, if we throw the same coin, we cannot say whether heads or tails will show up.

Therefore, we can only say that an event must take place if we know all conditions and surroundings influencing it. A random result is when we know only a part of the conditions and surroundings affecting it. Mathematics or, more exactly, the calculus of probabilities, does not study individual simple random results but rather only mass events by studying a larger number of them. This is understood as events which occur more often or which may occur more often under certain and constant conditions. For example, we can throw a coin as many times as we like.

Relating to the study of a series of occurring events, we can say that each event will occur roughly as often, regardless of how often the attempt is repeated or how extensive the attempt is. For example, it does not matter whether we will throw a coin a hundred or a thousand times - heads and tails will show up roughly the same amount of relative times. The truth of this rule can be verified by a series of attempts. Each of us knows that, when flipping a coin, the number of fallen heads should be equal to the number of fallen tails. In this case, the relative occurrence of one possibility is equal to the relative occurrence of the other, which is 1/2.

The probability of a certain event occurring can be approximated experimentally using random experiments, this while the relative frequency of one of the events is set during the attempts. If we stay at our dice flipping example, the experiment would involve throwing the coin n times and recording every time a head falls up. The relative frequency is therefore given by the fraction m/n, where m represents the number of throws where a head showed up. Concerning the size of n, we can say that the relative frequency should be close to the actual probability. From the so-called "black law of large numbers", it ensues that, as n becomes larger, the relative frequency should more closely approach the actual probability.

If we perform a similar attempt with a die, we should observe that each number should fall about the same amount of times and correspond to 1/6 of the attempts. The condition though is that we are throwing a normal, six-sided die and that the attempts are random. The relative probability of each number falling is therefore, in this case, 1/6.

Whoever understands the basic rules of the calculus of probabilities will be able to calculate chance and perhaps be able to make a good decision when playing games or in their professional career. In a certain game where one throws five dice and tries to obtain a certain series of numbers with a certain combination, it is a great advantage to be able to calculate the probability that a certain event will occur. For example: what is the probability that, during two attempts (throws) with one die, the numbers 1 and 2 will fall, or otherwise two 4s or two 5s.

Mathematics studies only mass phenomena, where a number of P(A) cases occurs with event "A", when it actually occurred and which also expresses the relative probability. We call the number P(A) as the probability of event A. If, during our attempt, we label the number of different possibilities as n and the number of cases when the event A occurred as k, then the probability of A is calculated as follows: P = k/n, which is the share of successful attempts k to the total number of events. Using this formula is very simple if we know n and k. If these values are not known, they must be calculated from known information using combinatory to enable us to use the formula.

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