(expert opinion, educated guess,not based on numbers/ not really any data) Show Assigning probabilities based on judgement. -Is most appropriate when one can not realistically assume that the experimental outcomes are equally likely and when little relevant data are available. B/c subjective probability expresses a person's degree of belief, it is personal. Using the subjective method, different people can be expected to assign different probabilities to the same experimental outcome. Sample Space and ProbabilitiesSample Space and Experimental Outcomes We can use the letter \(S\) or \(\Omega\) to denote sample space. Supose we have a set of \(n\) experimental outcomes: \[S = \left\{ E_1, E_2, ..., E_{n-1}, E_n\right\}\] We can call \(S\) a sample space if:
Thinking about the world in terms of sample space is pretty amazing. Assigning Probability to Experimental Outcomes We can assign probabilities to events of a sample space. Since experimental outcomes are themselves events as well, we can assign probabilities to each experimental outcome. For the mutually exclusive and jointly exhaustive experimental outcomes of the sample space, there are two equirements for assigning probabilities: - Each element of the sample space can not have negative probability of happening, and also can not have more than \(1\) probability of happening, with \(P\) denotes probability, we have: \[0 \le P(E_i) \le 1\] - The probabilities of all the mutually exclusive and jointly exhaustive experimental outcomes in the sample space sum up to \(1\). For an experimental with \(n\) experimental outcomes: \[\Sigma_{i=1}^{n} P(E_i) = 1\]
Union and Intersection and ComplementsDefinitions:
Probabilities for Complements and Union The Probabilities of Complements add up to 1: \[P(A) + P(A^c) = 1\] The Addition Law: \[P (A \cup B) = P(A) + P(B) - P (A \cap B)\] If two events \(A\) and \(B\) are mutually exclusive, which means they do not share any experimental outcomes (sample points), then: \(P (A \cap B) = 0\), and \(P (A \cup B) = P(A) + P(B)\). The Multiplication Law for Indepedent Events: \[P (A \cap B) = P(A) \cdot P(B)\] If the probability of event \(A\) happening does not change the probability of event \(B\) happening, and vice-versa. The two events are independent. Below we arrive this formulation from conditional probability. Conditional ProbabilityWe use a straight line \(\mid\) to denote conditional probability. Given \(A\) happens, what is the probability of \(B\) happening? \[P (A \mid B) = \frac{P(A \cap B)}{P(B)}\] This says the probability of \(A\) happening given that \(B\) happens is equal to the ratio of the probability that both \(A\) and \(B\) happen divided by the probability of \(B\) happening. The formula also means that the probability that both \(A\) and \(B\) happens is equal to the probability that \(B\) happens times the probability that \(A\) happens conditional on \(B\) happening: \[ P(A \cap B) = P (A \mid B)\cdot P(B)\] If \(A\) and \(B\) are independent, that means the probability of \(A\) happening does not change whether \(B\) happens or not, then, \(P (A \mid B) = P(A)\), and: \[ \text{If A and B are independent: } P(A \cap B) = P(A) \cdot P(B)\] This is what we wrote down earlier as well. What is the outcome of an experiment called?A result of an experiment is called an outcome. The sample space of an experiment is the set of all possible outcomes.
What is a list of possible outcomes called?The list of all the possible outcomes is called the SAMPLE SPACE (S). An event is any outcome or set of outcomes of a random phenomenon. An event must be present in the sample space.
What is outcome according to probability?In probability theory, an outcome is a possible result of an experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment).
What is the set of all the possible outcomes of an event called?The set of all possible outcomes of an event is essentially called a sample space and the events are the subsets of this sample space.
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