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In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event be 'I have a new phone'; event be 'I have a new watch'; and event be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy.
Alternatively, if a person is tested as positive for dengue fever, they may have only a 15% chance of actually having this rare disease due to high false positive rates. In this case, the probability of the event B (having dengue) given that the event A (testing positive) has occurred is 15% or P(B|A) = 15%.
Then the unconditional probability that = is 3/6 = 1/2 (since there are six possible rolls of the dice, of which three are even), whereas the probability that = conditional on = is 1/3 (since there are three possible prime number rolls—2, 3, and 5—of which one is even).
If the points in the joint probability distribution of X and Y that receive positive probability tend to fall along a line of positive (or negative) slope, ρ XY is near +1 (or −1). If ρ XY equals +1 or −1, it can be shown that the points in the joint probability distribution that receive positive probability fall exactly along a straight ...
Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
In probability theory, the chain rule [1] (also called the general product rule [2] [3]) describes how to calculate the probability of the intersection of, not necessarily independent, events or the joint distribution of random variables respectively, using conditional probabilities.
Philosophers including Robert Stalnaker argued that ideally, a conditional event algebra, or CEA, would support a probability function that meets three conditions: 1. The probability function satisfies the usual axioms. 2. For any two ordinary events A and B, if P(A) > 0, then P(A → B) = P(B | A) = P(A ∧ B) / P(A). 3. For ordinary event A ...
In the latter two examples the law of total probability is irrelevant, since only a single event (the condition) is given. By contrast, in the example above the law of total probability applies, since the event X = 0.5 is included into a family of events X = x where x runs over (−1,1), and these events are a partition of the probability space.