Independent Events Probability
Independent events in probability refer to two or more events that do not affect each other's outcomes. In other words, the occurrence of one event does not influence the probability of the other event happening. When events are independent, the probability of both events occurring can be found by multiplying their individual probabilities. This concept is fundamental in probability theory and has applications in various fields.
8 Key excerpts on "Independent Events Probability"
eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...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. If a coin is tossed and it lands as heads, this outcome does not influence the outcome of the next toss, which can be either heads or tails with the same probability as in any of the previous tosses. The probability of the joint occurrence of two independent events is given by: P (A ∩ B) = P (A and B) = P (A) × P (B). More generally P(A ∩ B ∩ … ∩ N) = P(A) × P(B) × … × P(N), and this is called the multiplication rule of probability, which is valid for equally and unequally likely events. For example, what is the probability of obtaining four consecutive “1”s when rolling a die? This is given by: P(four “1” s) = P(“1”) × P(“1”) × P(“1”) × P(“1”) = (1 6) 4 = 1 1296. As a further example, assuming that in a given population sex and hair colour (dyed hair is not valid!) are independent events with P(Female) = 0.5 and P(Blond hair) = 0.3, then P (Female and having blond hair) = P (Female) × P (Blond hair) = 0. 5 × 0. 3 = 0.15. 3.5 Probability trees Independent events, either with the same or different probability of occurrence, are often combined. It is then useful to see what happens in these cases and what are the probabilities of each of the obtained events. Let us illustrate what we mean using a couple of examples in which independent events are considered. In the first example, we want to calculate the probability of each of the possible outcomes obtained when a fair coin is tossed three times. A tree is used to calculate the probability of occurrence of each of these events (see Figure 3.5). Each branch of the tree gives the probability of an outcome. Following each branch tells us the probability of each set of three possible tosses...
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...In other words, two events A and B are statistically independent if: P A a n d B = P A × P B Independent events can also be of the form: P A | B = P A (a n d) P B | A = P B So the probability of A given B is the same as the probability of only A happening, since A and B are independent of one another. Similarly, the probability of B given A is the same as the probability of only B happening, because they are independent events. Any finite subset of events A i 1, A i 2, A i 3, …, A i n are said to be independent, if: P A i 1, A i 2, A i 3, …, A i n = P A i 1 P A i 2 P A i 3 … P A i n A typical example of independent events is flipping a coin twice, the outcome of head (H) or tail (T) facing up in the first toss does not affect the outcome in the second toss. Suppose a bank manager discovered that the probability that a customer prefers to open a saving account is 0.80. If three customers are selected at random, the probability that they will prefer saving accounts are independent events, since their choices of account do not affect one another. However, two events A and B said to be dependent events if occurrence of event A affects the occurrence of event B or vice versa. The probability that B will occur given that A has occurred is referred to as conditional probability of B given A, and it can be written as P B | A. Example 4.7: A company is organizing a project team from 3 departments (the administrative department, the marketing department and the accounting department) with a total of 30 employees. There are 8 employees are in the administrative department, the marketing department has 12 employees and the accounting department has 10 employees...
- George V Chilingar, Leonid F. Khilyuk, Herman H. Reike(Authors)
- 2012(Publication Date)
- Gulf Publishing Company(Publisher)
...706), one needs to evaluate the probability that water in the river is unpolluted and healthy provided that it has a high ability for self-cleaning. In formal notations, one needs to find the conditional probability P (H 4 / S). Using information from Fig. 5.1, one can formalize the events of interest in the following way: and Identifying probabilities of these events with their relative frequencies, one obtains Conditional probabilities are defined by corresponding relative frequencies: According to the formula of total probability (Eq. 5.5), On substitution of the corresponding numbers, Application of Bayes’ formula yields and INDEPENDENCE OF EVENTS Let A and B be two events of the same experiment and P(B) > 0. Event A does not depend (stochastically) on event B if It is noteworthy that if A does not depend on B and P (A) > 0, then B does not depend on A, because, according to Bayes’ formula, (5.8) Using independence of A and B, one obtains Thus, one can formulate the following definition. Definition 5.2. Two events A and B are called (stochastically) independent if one of them does not depend on another or has probability of zero. The second part of the definition is, of course, a convenient complement. Two events A and B are independent of each other if and only if (5.9) Assume that A and B are independent. If P (B) > 0, then If P (B) = 0, then 0 ≤ P (AB) ≤ P (B) = 0 ⇒ P(AB) = 0 (because AB ⊂ B). But P (A) P (B) is also equal to zero. Therefore, Eq. 5.9 is true. Conversely, assume that Eq. 5.9 is true and P (B) ≠ 0. Then This shows independence of events A and B. If P (B) = 0, then independence is implied by the above definition. Properties of independent events If two events A and B are independent, then the following pairs of events are also independent: A c and B, A and B c, A c and B c. It is sufficient to prove these properties just for any one of these pairs...
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...8 Probability 8.1 Introduction The concept of probability was introduced late in the seventeenth century. This concept was introduced in problems relating to games of probability (i.e., tossing a coin, playing cards). But the probability concept is now used in almost all areas of study such as economics, statistics, industry, engineering, and business. Probability is related to the study of events that are going to happen or not. Before going further, let’s define some of the basic terms that are going to be used in the definition of probability. 8.2 Definitions for Certain Key Terms 8.2.1 Experiment An experiment means an activity or measurement that result in an outcome. Example : Tossing a single coin for 50 times. 8.2.2 Sample Space Sample space refers to the collection of all possible events of an experiment, denoted by S. Example : In a coin-tossing experiment, the sample space should contain the possible outcomes of a head (H) or a tail (T); S = {H, T} 8.2.3 Event Event means one or more of the possible outcomes of an experiment; it is a subset of a sample space. Example : In throwing a dice, S = {1, 2, 3, 4, 5, 6} contains the face 1 is an event. 8.2.4 Equally Likely Events In a sample space containing at least 2 events, the chance of the occurrence of each of the event is equal. Example : In a coin-tossing experiment, having a head or tail in a trial is equal to ½ each or 50%. 8.2.5 Mutually Exclusive Events Events are said to be mutually exclusive if the outcome is only 1 element at a time. There is no chance that 2 or more events happen together. Alternatively, it is called an ‘incompatible event’. Example : In a coin-tossing experiment, we can have either head or tail as an outcome...
eBook - ePub- Roy Weatherford(Author)
- 2022(Publication Date)
- Routledge(Publisher)
...But evidently not everyone falls in these exalted categories, for consider these recent definitions: Probability. The ratio of the number of ways in which an event can occur in a specified form to the total number of ways in which the event can occur. 104 Probability, Mathematical. If an event can happen in a ways and fail in b ways, and, except for the numerical difference between a and b, is as likely to happen as to fail, the mathematical probability of its happening is a /(a + b) and of its failing, b /(a + b). 105 The Classical definition lives! And as long as human beings continue to face situations where the outcome is unknown but the alternatives are all felt to have an equal chance of occurrence – as long, in short, as we continue to gamble at dice and cards – the Classical Theory of Probability will continue to be the working theory of the ordinary person....
- A. Hasofer, V.R. Beck, I.D. Bennetts(Authors)
- 2006(Publication Date)
- Routledge(Publisher)
...Then, using the theorem of total probability, we can evaluate P (D) as follows: 3.6 The concept of independence Intuitively, if event B has no effect on event A we can say that A is independent of B. More precisely, if knowledge that event B has occurred does not affect the probability of A, we say that A is independent of B. This is expressed mathematically as: Replacing P (A | B) by its expression from Definition 3.4.1, we find: This can be rewritten P (A ∩ B) = P (A) P (B). It is interesting to note that the last expression implies that In other words, if A is independent of B, then B is independent of A. Thus, we can say that if P (A ∩ B) = P (A) P (B) then A and B are independent. This definition can be extended to more than two events. We say that A 1, A 2,. . . , A n are independent if Example 1. Returning to the example in Section 3.4, we can ask whether the event “window open” is independent of the event “door open”. We have But Thus, we see that in the situation considered knowing that the door is open decreases the probability that the window will be open, and the two events are not independent. 2. Suppose that we know that the probability that a smoke detector is defective is 0.05. Consider two smoke detectors in two separate rooms. We can reasonably assume that the two events “smoke detector 1 is defective” and “smoke detector 2 is defective” are independent, if they have been independently bought and independently installed. We can then state that the probability that both smoke detectors are defective is 0.05 × 0.05 = 0.0025. This result is the key to understanding the principle of redundancy in system design. Suppose we install several components to carry out the same function in some system. Suppose further that the components have independent probabilities of failure...
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
...In such a case we denote the event B as A c. The set B consists of exactly those simple events which are in S but not in A. • Partition: A collection of events A 1, A 2, …, A k forms a partition of the sample space if the collection is exhaustive and the events in the collection are mutually exclusive. The simplest partition is provided by two events that are complements of each other. The events of buying coffees of different brands (see Case Study 3.3) partition the corresponding sample space. We denote by Pr (A) the probability of an event A. It is restricted to lie in the interval [0,1], with the interpretation that Pr (A) = 1 represents a sure event, while Pr (A) = 0 implies that the event A will never occur. With this background, we are now ready to present the Classical Definition of Probability. Assume that we have performed a statistical experiment for which 1. The total number of possible simple events, N, is finite, and 2. Each simple event is equally likely. Then the probability of any event A is defined as Pr (A) = N (A) / N (S) (5.3) where N (S) = N is the total number of simple events and N (A) is the number of simple events favorable to A. Consider, for instance, the experiment of tossing a fair coin. The classical definition will indicate that the probability of getting a head for this experiment is 1/2. Let us discuss this definition further with this particular example in mind. While the probability of getting a head is precisely defined by the classical definition, there is still a philosophical abstraction about it. Tossing the coin ten times does not necessarily lead to exactly five heads. After all, it is this uncertainty which is the basis of the theory of probability. However, as it turns out, within the fold of this uncertainty, the outcomes of the random experiments begin to show some regular behavior when the number of experiments keeps getting larger and larger...
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)
...Consider a deck of cards and let event A be the drawing of a heart and event B the drawing of a black card. This time there is no overlap since a card cannot satisfy both outcomes. This is illustrated in Figure 2.2. Figure 2.2 Mutually exclusive events The previous theorem does not apply here since the probability of both event A and event B taking place is zero. It follows that the probability of drawing either a heart or a black card (event A or event B) is P a + P b and for multiple events, the probability of observing either event A, or event B, or event C, or event D, etc., is P a + P b + P c + P d e t c In the playing card example Pa = 0.25 Pb = 0.5 and therefore the probability of drawing either a heart or a black card is 0.25 + 0.5 = 0.75. Exercise 1 Manipulating Probabilities Assume that: Having dark hair is 80% likely. Having blue eyes is 25% likely. Being bald is 5% likely. Being taller than 1.8 m is 50% likely. What is the probability of the following: Having dark hair and blue eyes? Having dark hair or blue eyes? Having dark hair and being bald? Having dark hair and blue eyes and being taller than 1.8 m? Having dark hair or blue eyes or being taller than 1.8 m? Having dark hair and blue eyes, or being bald? Having neither blue eyes nor not being bald? Conditional Probabilities The following example illustrates a state of affairs in which the probability of an event is conditional on the probability of some other factor. Assume that a person, at random, has a 1% probability of suffering from a particular form of cancer (in other words 1% of the population are known to suffer). Assume, as is often the case, that there is a test to determine if the cancer is present but that the test is not perfect. Like many tests, it may return a false positive despite the subject not suffering from the condition and it might also return a false negative in that it fails to detect a real case...
