# PROBABILITY

Probability is the branch of mathematics concerned with some experiments called random experiments in which it is known all the possible results, but there is no a certainty of what will be a specific result. For example, random probability experiments are tossing a coin, tossing a six-sided die, drawing a card from a standard deck of 52 plating cards.

The examples used above correspond to what is called the mathematical or theoretical probability, that must be distinguished from the statistical or empirical probability.

Some Probability Definitions.-

Sample Space.- Associated to a random experiment, it is called Sample Space (S) to the set of all the possible results for that experiment.

When tossing a coin, the sample space is S={landing head, landing tail} or S={h, t}

When tossing a die, the sample space is S={landing 1, landing 2, landing 3, landing 4, landing 5, landing 6} or S={1, 2, 3, 4, 5, 6}.

When tossing two coins, the sample space is S={(h,h), (h,t), (t,h), (t,t)}

When tossing three coins, the sample space is S={(h,h,h), (h,h,t),(h,t,h),(h,t,t),(t,h,h),(t,h,t),(t,t,h),(t,t,t)}

Event.- An event is any subset of the sample space. For example, in the sample space S={1,2,3,4,5,6} when tossing a die, examples of events are:

1. To get a prime number, A={2,3,5}
2. To get a prime number and even, B={2}
3. To get a number equal or greater than 5,C={5,6}

Mutually Exclusive Events.- Two or more events are mutually exclusive if not more than one of them can occur in a single trial, that is if and only if its intersection is empty. For example, when tossing a die the events B={2} and C={5,6} are mutually exclusive because their intersection is empty.

Complementary Events.- If AB = and AB=S, it is said that A and B are complementary events:
Ac=B and Bc=A

Mathematical Probability.- If in a random experiment all the results have the same probability, that is, the occurrence of one result is equally possible than the occurrence of any of the other results, then the probability of an event A is the ratio:
P(A)=number of cases favorable to A/ total number of cases

From this definition, the probabilities of the possible results of the experiment can be determined without putting into effect the experiment.

From the definition is deduced that:
0 P(A) 1 The probability is a real number between 0 y 1, inclusive, or 0% P(A) 100% in percentage. P() = 0 and P(S) =1

Statistical or Empirical Probability.- The relative frequency of the A result of an experiment is the rate
FR=number of times A occurs/ number of times the experiment is put into effect

If the experiment is repeated a great number of times, The FR value will approach to the probability P of the event. For example, if I toss 100 times a coin, the number of times I get head is nearly 50, that is FR is close to 50%.

Probability Sets
Mathematics, Help Scope