Probability Conditional event

The conditional probability of event B given A is denoted by P(B A) and defined as... 

Tlie conditional probability of event B, no failures in 10 years, given tliat tlie failure rate is Z per year, is obtained by applying the Poisson distribution to give... 

It is often important to be able to extend our present notion of conditional probability to the case where the conditioning event has probability zero. An example of such a situation arises when we observe a time function X and ask the question, given that the value of X at some instant is x, what is the probability that the value of X r seconds in the future will be in the interval [a,6] As long as the first order probability density of X does not have a Dirac delta function at point x, P X(t) = x = 0 and our present definition of conditional probability is inapplicable. (The reader should verify that the definition, Eq. (3-159), reduces to the indeterminate form in this case.)... 

The adverse event for a patient may indicate a medical condition such as hypercholestimia, so there may be a request to ensure that there are elevated cholesterol laboratory data to verify such a claim. You can sometimes make this kind of verification with programming if you know precisely which lab tests are involved and what level indicates a probable adverse event. 

Event trees are used to perform postrelease frequency analysis. Event trees are pictorial representations of logic models or truth tables. Their foundation is based on logic theory. The frequency of n outcomes is defined as the product of the initiating event frequency and all succeeding conditional event probabilities leading to that outcome. The process is similar to fault tree analysis, but in reverse. 

Let us introduce the concept of conditional probability. We denote the joint probability of events A and B by p(A,B) and the conditional probability of occurrence of event A given event B by p(A B). Thus, we have... 

We denote the conditional probability that event A occurs, given that event B does not, by p(A B). Using the same prescription as that of Eq. (206), we should write... 

The most natural way to introduce aging is through the age-specific failure rate illustrated in the nice book of Cox [94]. To apply Cox s arguments to the BQD physics, we have to identify the failure of a component, for instance an electric light bulb, with the occurrence of an event, as discussed in Section X. According to Cox, let us consider a sequence of the Gibbs ensemble of Section X known not to have produced an event at time t and let r(t) be the limit of the ratio to At of the probability of event occurrence in [t, f+A]. In the usual notation for conditional probability,... 

Often the probability of an event depends on one or more related events or conditions. Such probabilities are called conditional We will write p A B) for the probability of event A given B (or the probability density of A given B if the sample space of A is continuous). 

It is also possible to state the probability of an event. A, as a function of two or more conditional events. If the events B and C are mutually exclusive and exhaustive - for example, they represent male and female - the probability of event A can be expressed as ... 

Recall also from Chapter 6 that the marginal probability of an event can be expressed as a series of conditional probabilities as long as the conditional events are mutually exclusive and exhaustive. This allows us to express the probability. 

The recursive approach uses an elementary law of conditional expectation. Let A be an event and A its complement. Let Y be a random variable, E Y) its expectation (or average value) and E Y A) is conditional expectation, given that the event A has occurred. P A) is the probability that event A occurs. Then the law of total probability for expectation is [31] ... 

Consider two events, A and B. Suppose that an event B has occurred. This occurrence may change the probability of A. We denote this by P A B), the conditional probability of event A given that B has occurred. 

Use A and B to denote two events. The conditional probability of event A provided the occurrence of event B is given by ... 

For the Wiener process, we know that W(f) — W(5 ) is normally distributed with mean zero and variance f - this does not depend on whether there is additional information available regarding its value prior to min(f, s), that is to say, the Wiener process is a Markov process. More generally, when we say that a stochastic process is Markovian, we mean that the probability of a future event conditioned on the current state of the process and the past history of the process is the same as the probability conditioned on the current state of the process. Let A x), B(x) and Q(x) be three events dependent on state variable x, and let t+ > to > t- be three times, then, for a Markov process X(f) we have (using the notation for conditional probability) ... 

The conditional probability Pr(yi B) is the probability of event A, given event It is defined by... 

Normally, the probability of event A would be given by the area of circle A divided by the total area. Conditional probability is different. For example, consider the situation when Event B has occurred. This means that only the state space within the area of circle B needs to be examined. This is a substantially reduced area The desired probability is the area of circle A within circle B, divided by the area of circle B, expressed by ... 

This states that the probability of event A equals the conditional probability of A, given that B has occurred, plus the conditional probability of A, given that B has not occurred. This is known as Bayes rule. It is used in many aspects of reliability engineering. 

This survey of IT probes of chemical bonds continues with some rudiments on the entropic characteristics of the dependent probability distributions and information descriptors of a transmission of signals in communication systems [3,4,7,8]. For two mutually dependent (discrete) probability vectors of two separate sets of events a and b, P(a) = P(fl) =Pi) =p and P(b) = P(pp = qj = q, one decomposes the joint probabilities of the simultaneous events a/ b = [a,Air) into these two schemes, P afdb) = P(fli/ b) = Jt,y] = 31, as products of the margin probabilities of events in one set, say a, and the corresponding conditional probabilities P( la) = [P(/li) = 3t,y/Pj] of outcomes in set b, given that events a have already occurred [jty=p (j i). The relevant normalization conditions for the joint and conditional probabilities then read ... 

In the examples used above, the events described were mutually exclusive, that is, the fact that a coin comes down heads on one throw has no influence on what it will come down on the second throw, one man aged 50-55 dying has no influence on the death of others of the same age, and an accident on one shift does not make accidents on other shifts more or less likely. However, there are events which do influence each other, and where this is the case, the third type of probability, conditional probability, applies. These probabilities express, in numerical terms, the chance of event A occurring, given that event B, which will influence it, may or may not happen. 

Conditional probability, and the related topic of independence, give probability a unique character. As aheady mentioned, mathematical probability can be presented as a topic in measure theory and general integration. The idea of conditional probabihty distinguishes it from the rest of mathematical analysis. Logically, independence is a consequence of conditional probability, but the exposition to follow attempts an intuitive explanation of independence before introducing conditional events. 

The change in the population during t to t + h depends upon the foregoing events. For the average change, we need their respective probabilities conditional on A. These have already been defined and are recalled as... 

The probability that event i is first is p(i). Then the conditional probability that event j is second is p(j)lii - pii)). The joint probability that i is first, j is second, and k is third is... 

Consideration should be given in the safety analysis to the possible duration of extreme events, particularly for extreme weather conditions. ITius if the extreme conditions postulated for the site could endure for a considerable period, the feasibility of providing any backup measure from off the site should be evaluated, in view of the damage that is likely to occur and the probable conditions for the emergency services. Therefore, realistic assessments should be made of the ability to respond off the site under extreme conditions in the site region, when other demands for emergency services may be paramount. Either an adequate capacity should be provided for such circumstances or such backup measures should be excluded from the safety analysis. 

Law of Total Probability n Also known as the Theorem on Total Probability or the Law of Alternatives. It states that the probability, P(A) of an event, A, is equal to the sum of the conditional probabilities of A, given events, Ej, P(A 1 ), times the probability of event, , for i = 1, 2, 3,. ..,N where Nisa. positive integer or infinity, and where Efi are non-overlapping and form a partition of a sample space that covers the sample space of A. This can be expressed as ... 

A component failure is a dependent failure when a conditional relationship exists between two components, whereby the failure of one is conditional upon failure of the other the second failure depends on the first failure occurring. In probability theory, events are dependent when the outcome of one event directly affects or influences the outcome of a second event. To find the probability of two dependent events both occurring, multiply the probability of A and the probability of B after A occurs P(A and B) = P(A) P(B given A) = P(A) P(BIA). This is known as conditional probability. Two failure events A and B are said to be dependent if P(A and B) P(A)P(B). In the presence of dependencies, often, but not always, P(A and B) > P(A)P(B).This increased probability of two (or more) events is the reason dependent failures are of safety concern. It should also be noted that when redundant dependent components are used in a system, they are both susceptible to the same dependent failure, which creates a CCF dependency. 

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