Events, that is, particular outcomes like rolling a 12 with two dice, are important both inside math and outside it. The obvious relate to games of chance, anything from a game 3 Card Monte to a game of Roulette. On the other hand, businesses such as insurance companies need to know about events concerned with car accidents, death, and footballers having accidents. To decide whether it is worth taking some action, and what action to take, we rely on a measure of the likelihood of an event that we call probability. For reasons the size of the probability of an event is expressed as a number between zero and one or a percentage less than one 100.
Probabilities can be determined in two ways theoretically or experimentally. Many simple events can be found theoretically. The probability of getting a 4, on the roll of a dice can be found theoretically. First you have to know the set of all possible outcomes. In the case of a dice this is {1, 2, 3, 4, 5, 6}. So there are six events in the event space. There is only one event in this event space that produces a 4. The probability of getting a 4 when you roll a dice is 1/6, the 1 is for the number of times a four can occur among all possible events and the 6 is for the number of all possible events (the size of the event space). When calculating probabilities theoretically, the probability is given by the equation
As another example consider what happens when you roll two dice. What are the chances of getting a 4 then? Well here the event space is of size 36 {1, 1), (1, 2), (1, 3), (6, 4), (6, 5), (6, 6} and the times that 4 is possible are given by {1, 3), (2, 2), (3, 1)}. The probability of getting a 4 is 3/36 = 1/12.
But not all events can have their probabilities calculated theoretically. The probability of there being three major earthquakes in a year or the chances that it will rain tomorrow. We cant produce an event space to measure these probabilities by. So we have to take measurements over a long period. In that time we can compare the number of rainy days over the number of days altogether - the number of favourable outcomes (yes, a rainy day is favourable in this context) over the number of possible outcomes. So in a case like this, where we have to calculate probability experimentally, the probability is given by
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