Mutually exclusive events

PREPROCESSOR - Modifies the FTAP punch file output for common cause and dependent analyses (conditional probabilities), removes complemented events, and corrects for mutually exclusive events. 

Eliminating min cut sets with mutually exclusive events. 

Conducting common cause and dependent event analysis Dropping complemented events and performing the subsequent minimization Generating block files (i.e., a set of Boolean equations) for subsystems Eliminating mincutsets with mutually exclusive events... 

Note that tlie union of A and B consists of tliree mutually exclusive events AB, AB, AB. 

Consider n mutually exclusive events A, A2,. .., A whose union is tlie sample space S. Let B be any given event. Then Bayes theorem states... 

MUTUALLY EXCLUSIVE EVENTS events that cannot occur simultaneously. For example, when a die is rolled, a five can be rolled, or a six can be rolled, but both a five and a six cannot be rolled simultaneously. So rolling a five and rolling a six are mutually exclusive. However, in a deck of standard playing cards, when a card is chosen, the card can be a spade, or the card can be a queen. A spade and a queen can be chosen, so choosing a spade and choosing a queen are NOT mutually exclusive. 

Mutually exclusive events cannot happen simultaneously. 

To find the probability that one of two or more mutually exclusive events will occur, add the probabilities of each event occurring. For example, in the previous problem, if we wanted to find the probability of drawing either a green or black button, we would add the probabilities together. 

To obtain equations for the state probabilities, write the equation for the state probability at t + At as the sum of joint probabilities for all the mutually exclusive events that enumerate all the possible ways in which a particle starting in i at 0 could pass through the various compartments at time t AND end up in j at t + At. These joint probabilities can be expressed as the product of a marginal by a conditional probability. The state probability p (t) that a given particle starting in i at time 0 is resident in compartment s at time t plays the role of the marginal and the transfer probability hSj (t) At that a given particle resident in compartment s at time t will next transfer to compartment j, i.e., at time t + At plays the role of the conditional probability... 

Consider the irreversible two-compartment model with survival, distribution, and density functions starting time, the molecules are present only in the first compartment. The state probability p (t) that a molecule is in compartment 1 at time t is state probability p2 (/,) that a molecule survives in compartment 2 after time t depends on the length of the time interval a between entry and the 1 to 2 transition, and the interval I, a between this event and departure from the system. To evaluate this probability, consider the partition 0 = ai < a.2 < < o.n 1 < an = t and the n — 1 mutually exclusive events that the molecule leaves the compartment 1 between the time instants a, i and a,. By applying the total probability theorem (cf. Appendix D), p2 (t) is expressed as... 

Total probability theorem. Given n mutually exclusive events A, ..., An, whose probabilities sum to unity, then... 

Consider n mutually exclusive events Ai, A2.A whose union is tlie sample... 

But there are 1, 2, 3,... y/of the elements exceeding the value. Also, for mutually exclusive events, the intersection probability is equal to zero. Thus,... 

Substituting Equation (1.10) into Equation (1.9), assuming mutually exclusive events. 

Mutually exclusive events—Events that cannot occur at the same time. 

For a discrete sample space of mutually exclusive events Ei, one defines a corresponding set of event probabilities pi = p Ei), which may be given or may be estimated from observations. The Pi have the properties... 

In calculating the probability of observing an event Sk, the conditional probability is applied in the following way. Suppose Si, S2, S3,... is full set of mutually exclusive events independent of each other. By full set is meant that it includes all possible events and that the events S, S2, S3,. .. may always be observed, but not simultaneously. If Sk is known to be dependent on Si, S2, S3,. .., then we can find prob Sk by using the total probability formula [5, p.27] where Z is the total number of events it reads ... 

In the special case that the events A and B cannot occur at the same time, they are said to be mutually exclusive, meaning that P(A and B) = 0. Hence, for mutually exclusive events A and B ... 

The probability of selecting a male at random can be calculated by adding the probabilities for the events "male < 45 years," "male 46-64 years," and "male > 65 years," because these are all mutually exclusive events. The probability can be calculated as follows ... 

The area under the curve is 1.0. This can be demonstrated formally using integral calculus, which is beyond the scope of this book A simpler demonstration is provided by Turner (2007, pp 94-5). That the area under the curve is equal to 1 is analogous to the statement that the probability of all mutually exclusive events must sum to 1. 

Figure 11-1. Venn Diagram Representations of Mutually Exclusive and Non-Mutually Exclusive Events...

E. Brambilla, Mdm2 overexpression and pl4ARF inactivation are two mutually exclusive events in primary human lung tumors, Oncogene 2002, 21, 2750-2761. 

Mutually exclusive events. Events which stand in that relationship to each other whereby one or other may occur separately but both cannot occur together. For such events the addition rule of probability is simpler than otherwise. 

Disjoint If A and B have no outcomes in common, that is, A n B = 0, where 0 is an empty set, then A and B are called disjoint or mutually exclusive events. 

This result can be used to update the prior probabilities of mutually exclusive events B, in light of the new information that A has occurred. The following example... 

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