Probability

Quantitative Results

Probability is a value that specifies whether or not an event is likely to happen. The value generally lies between zero to one. If an event comes out to be zero, then that event would be considered successful. If an event comes out to be one, then that event would be considered a failure.

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A sample space (S) is a non empty set whose elements are called outcomes. The events are nothing but the subsets of the sample space.

A probability space consists of the sample space and the function, which involves the mapping of the events to the real numbers in an interval of zero in such a way that the sample space is one. If A0 ,A1, ….. is the sequence of disjointed events, then the probability of the union of the sequence will be equal to the sum of all the disjointed events.

Conditional probability is the type that denotes a particular event when it is given that another particular event has occurred, provided that the occurrence of the other particular event is not equal to zero.

There is a product rule in probability that states that the intersection of any two particular events is equal to the product between the probability of the second event and the conditional probability of the events.

The theorem of total probability states that if the sample space is the disjointed union of events, for example B1, B2, …. then for all events of A, then the probability of A will be equal to the sum of the intersection between the event A and the disjointed events Bi.

Suppose the two events, A and B, have a positive probability. In this case, the event A would be independent of B if and only if the conditional probability of A given the events B is equal to the probability of A. It is important to remember that this independence probability would be applicable only when the probability of the event B would not be equal to zero.

There is also an independence product rule that states that the probability of the intersection of the two events is equal to the product of the event A and of the event B. It is important to remember that in the theory of probability, the disjointed events are not the same as that of the independent events.

The theory of probability is the logic of science. According to James Clerk Maxwell (1850), the true logic involves the calculus of probability, which takes into consideration the magnitude that is supposed to be reasonable.

The theory can be described with a popular example— the tossing of a coin with possible outcomes of “heads” or “tails.” Suppose “heads” is considered a success and “tails” is considered a failure. Thus, the probability of a success (“heads”) will be the probability of the value one, and the probability of failure (“tails”) is the value of zero. Similarly, rolling dice is another popular example.