Disjoint and Overlapping Events
Disjoint events in mathematics refer to events that cannot occur simultaneously, while overlapping events can occur at the same time. In probability theory, disjoint events have no outcomes in common, whereas overlapping events share common outcomes. Understanding the distinction between disjoint and overlapping events is important for calculating probabilities and making predictions in various mathematical scenarios.
5 Key excerpts on "Disjoint and Overlapping Events"
eBook - ePub- Lawrence S. Aft(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
...Similarly, evaluating a part as acceptable and evaluating it as defective are mutually exclusive events. Two events are said to be non-mutually exclusive when they share a common area of occurrence or can both happen at the same time. Within a population sample, being male and having blue eyes are not mutually exclusive events. There are some people who are males and have blue eyes. Elementary algebraic-set theory illustrates these probability concepts best. Figure 3.1 shows the mutually exclusive events A and B via a Venn diagram. Note that there is no overlap of the circles representing the two events. The non-mutually exclusive events A and C are shown in Figure 3.2. Notice the area of overlap. It represents the commonality of the events. Figure 3.1 Venn Diagram for Two Mutually Exclusive Events A and B Figure 3.2 Venn Diagram for Two Non-Mutually Exclusive Events A and B Two or more events are said to be independent if the occurrence of one event does not affect the occurrence of the second event in any way, shape, or form. Selection of one does not alter the chance of selection of the other. The probability of selecting a defective part from a large sample of bolts is not influenced by the probability of having selected a defective part from a preceding sample of bolts. Rules There are two basic rules for probability. These are the addition and the multiplication rules. The addition rule is used when the probability of one event or another event will occur. The multiplication rule is used when the probability of one event and another will occur. Addition Rule for Mutually Exclusive Events The probability of occurrence of event A or event B, when events A and B are mutually exclusive, is the sum of the probabilities of the occurrence of event A and the occurrence of event B...
eBook - ePub- Dušan Teodorović, Miloš Nikolić(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...Point 4 describes the situation where the complement of the complement of the event is the initial event. Point 5 states that the complement of the intersection of two events equals the union of their complements. Point 6 describes that the intersection between an event and its complement equals an impossible event. Point 7 states that the intersection between an event and a sure event is the original event. Two events, A and B, are mutually exclusive, if the occurrence of one event means nonoccurrence of the other. In other words, mutually exclusive events A and B cannot happen simultaneously (Figure 4.6). Figure 4.6 Mutually exclusive events Example 4.2 A container ship arrives at the port to have loading and unloading operations carried out. If all berths at the port are occupied, the ship has to wait until any one of the berths is going to become idle. The total time that the ship spends at the port consist of waiting time and service time. Let us denote with A an event that the total time which the ship spends at the port is less or equal to 10 hours. A mutually exclusive event of event A is that the total time the ship spends at the port is more than 10 hours. 4.2 Probability of a random event What is a probability ? Probability is a non-negative real number not greater than one. Could probability be equal to zero? Yes. Could probability be equal to one? Yes. Could probability be greater than one? No. How could we calculate the probability of a specific event? Usually, we repeat the experiment many times, and we count the number of trials m that describe our event. Let us denote by n the total number of trials. By performing the experiment, we observe that m trials out of n trials describe our event. We denote by P (A) the probability of event A...
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...Given that for two mutually exclusive events n (A ∩ B) = 0, it follows that the probability of the intersection of two mutually exclusive events is 0, therefore, the probability of their union is given by: P (A ∪ B) = P (A) + P (B). Figure 3.4 Venn diagram representing three mutually exclusive events A, B, and C. notice that there is no overlap between the circles, indicating that no simple event is common to any of the possible pairs of events. This is known as the additive rule of probability, and for N mutually exclusive events it generalises to P(A ∪ B ∪ … ∪ N) = P(A) + P(B) + … + P(N). From the additive rule it follows that, if two events are mutually exclusive and exhaustive (i.e., all the simple events in the sample space have to be included in the union of the two mutually exclusive events), then: P (A ∪ B) = 1, which for N mutually exclusive and exhaustive events generalises to: P (A ∪ B ∪ … ∪ N) = 1. Note that the. equations P (A ∪ B) = P (A) + P (B) − P (A ∩ B), P (A ∪ B ∪ … ∪ N) = P (A) + P (B) + … + P (N) P (A ∪ B ∪ … ∪ N) = 1 are valid when events are either equally likely to occur or unequally likely to occur (the last two equations only apply to mutually exclusive events). The multiplication rule For mutually exclusive events we know, by definition, that P(A ∩ B) = 0. Is there a formula to calculate P(A ∩ B) (or equivalently P(A and B)) for different types of events? Well, it turns out that there is a special formula to calculate the probability of the joint occurrence of two or more events provided that these events are independent. A set of events are said to be independent if the outcome of one event does not influence the outcome of any of the other events. Repeated tosses of a coin constitute independent events...
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...Mutually exclusive events A and B are denoted A ∩ B = ϕ. 4.5 Mutually Inclusive Events Two events said to be mutually inclusive, if there are some common outcomes in the two events. Getting a head and a tail when flipping two coins is an example of mutually inclusive events. Another example is getting an odd number or prime number when throwing two dice. 4.6 Venn Diagrams The use of Venn diagrams helps to visualize relations between sets. Figures 4.1 – 4.4 are diagrammatical representations of sets in a rectangular box. Figure 4.1: Venn diagram, A ⊂ S. Figure 4.2: Two sets A and B, where B ⊂ A. Figure 4.3: Two sets A and B, where A ∩ B. Figure 4.4: Two sets A and B, where A ∪ B is the shaded area. 4.7 Probability The probability of event A can be defined as the number of ways event A may occur divided by the total number of possible outcomes. It is mathematically defined as: P r o b a b i l i t y A = n u m b e r o f o u t c o m e s f a v o r a b l e t o A n u m b e r o f p o s s i b l e o u t c o m e s or (4.1) P A = n A n s Example 4.1: A fair die is rolled once, what. is B = 1, 3, 5 the probability of: (1) rolling an even number? Or (2) rolling an odd number? Solution P r o b a b i l i t y e v e n n u m b e r = n u m b e r o f e v e n n u m b e r n u m b e r o f p o s s i b l e o u t c o m e s Let A be a set of even. numbers, A = 2, 4, 6, and B be a set of odd numbers, B = 1, 3, 5, in tossing a die, with the sample space S = 1, 2, 3, 4, 5, 6. P A = 3 6 P B = 3 6 Example 4.2: What is the probability that an applicant’s resume will be reviewed within a week of submitting an application if 5,000 graduates applied for a job and the recruitment firm can only review 1,000 resumes in a. week? Solution P r o b a b i l i t y t r e a t i n g a r e s u m e = 1, 000 5, 000 = 0.2 4.7.1 Simulation of a Random Sample in R The R command sample is used to simulate drawing a sample...
eBook - ePubBasic Statistical Techniques for Medical and Other Professionals
A Course in Statistics to Assist in Interpreting Numerical Data
- David Smith(Author)
- 2021(Publication Date)
- Productivity Press(Publisher)
...The fact that the circles overlap represents the fact that the two events are not mutually exclusive – clearly it is possible to exceed a height of 5’6” and, also, to attain the age of 65 years. The area outside the circles represents all possibilities which are neither event A nor event B. Figure 2.1 Overlapping events The shaded (overlapping) portion of the diagram represents the possibility that both event A and event B will take place and the reader will doubtlessly recognise this as the situation to which the Multiplication rule applies. If Pa (the probability of event A occurring) is 0.8 If Pb (the probability of event B occurring) is 0.7 Then the probability that both event A and event B will occur is P a b = P a x P b = 0. 8 x 0. 7 = 0. 56 In other words, 0.8 proportion of the 0.7 proportion of the total number of trials will result in both A and B together. The rule can be extended to calculate the probability that all of n events will occur together. P a n = P a x P b x P c … … … x P n The above equations assume independence between the events Pa, Pb and so on. Consider the following case. The probability of a new-born cot death is approximately one in 8,000 births. Thus, the probability of two babies dying in the same family might be inferred to be one in 8,000 x 8,000 = one in 64 Million. This assumes that if a child in a family dies of infant death syndrome, then it is no more and no less likely that a subsequent child will die in that way. However, there are reasons why the two events might not be independent as, for example, a genetic factor, or an environmental factor unique to that family. A different calculation involves the probability of either A or B or both (in other words at least one) of the events occurring...
