Probability
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 results in zero, one would consider that event successful. If an event comes out to be one, one would consider that event a failure.
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The sample space (S) represents a non-empty set, and its elements represent outcomes. The events are nothing but the subsets of the sample space.
A probability space consists of a sample space and a function mapping events to real numbers. Here, the total probability is one. If A0, A1, … are disjoint events, the probability of their union equals the sum of their probabilities.
Conditional probability refers to an event given that another event has occurred, provided the other event’s probability is not zero. The product rule states that the intersection of two events equals the product of the second and the first event’s conditional probability.
The theorem of total probability states that if the sample space is a disjoint union of events (B1, B2, …). The probability of A is the sum of the probabilities of A and each Bi.
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Suppose the two events, A and B, have a positive probability. Event A is independent of B if P(A|B) equals P(A), provided P(B) is not 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.
One can describe the theory with a popular example—the tossing of a coin, which has possible outcomes of “heads” or “tails.” Suppose you consider “heads” a success and “tails” 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.
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