๐ Probability is an assignment of value to outcomes of experiments.

๐ข Outcome of an experiment can be finite, countably infinite, or uncountably infinite.

๐ก Events are sets of outcomes, and outcomes can be elementary events or subsets of events.

๐ Probability is defined based on the hypothesis that an infinite number of trials will approach a certain limit.

๐ฒ Probabilities are assigned to events based on a logical deduction system and must follow certain axioms.

๐ฏ The probability space is defined as the set of all possible outcomes, and events are subsets of this set.

๐ The probability of the union of two sets can be calculated by summing the probabilities of the individual sets.

๐ข Assigning numbers to sets in a way that satisfies the axioms of probability is crucial.

๐งฎ The relative frequency definition of probability is one way of assigning probabilities, but there are other possibilities.

๐ Probability of events occurring in sets A and B

๐ Defining probability for infinite sets and intervals

๐ Probability density functions and defining probabilities for subsets

๐ Probability theory involves assigning probabilities to events and understanding conditional probabilities.

๐ The alpha function in probability theory can be defined as 0 for all x less than -273, and can be extended beyond a finite range.

๐ Conditional probability is the probability of an event A given that another event M has occurred, and it is calculated using the formula: P(A|M) = P(A โฉ M) / P(M).

๐ Conditional probabilities utilize additional information to compute the probability of an event.

๐น If M is a subset of A, then probability of A given M is 1.

โ The total probability theorem states that the probability of an event can be computed using conditional probabilities.

๐ Partitioning of a set is when you have sets A 1 up to A n that are disjoint and their union is the entire set.

๐ The total probability theorem states that the probability of event B can be calculated by summing the probabilities of B given A i multiplied by the probabilities of A i.

๐ฌ Bayes theorem allows us to calculate the probability of A given B using the probability of B given A and the probabilities of A and B.

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