Mathematics
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.
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eBook - ePubStatistics for the Behavioural Sciences
An Introduction to Frequentist and Bayesian Approaches
- Riccardo Russo(Author)
- 2020(Publication Date)
- Routledge(Publisher)
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:
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, thenP(four “1” s) = P( “1” ) × P( “1” ) × P( “1” ) × P( “1” ) ==(41 6)1 1296.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. Since tosses are independent events, it follows that the probability of obtaining a series of three specific tosses is given by applying the multiplication rule of probability, i.e., by the product of the probability of each toss. Furthermore, notice that using the probability tree all possible triplets are obtained, and that these triplets represent mutually exclusive events. Therefore the sum of the probabilities of all possible triplets is one, i.e., P(HHH ∪ HHT ∪ … ∪ TTT
eBook - ePub- Mustapha Abiodun Akinkunmi(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
independent if the occurrence of event A does not affect the occurrence of event B. In other words, two events A and B are statistically independent if:Independent events can also be of the form:PA a n d B= P A × P BPA | B= P A ( a n d ) PB | A= P BSo 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 eventsare said to be independent, if:A,i 1A,i 2A, … ,i 3Ai nP= PA,i 1A,i 2A, … ,i 3Ai nAPi 1APi 2A… Pi 3Ai nA 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 asP.B | AExample 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. If two employees are selected to be on the team, one after the other:
- George V Chilingar, Leonid F. Khilyuk, Herman H. Reike(Authors)
- 2012(Publication Date)
- Gulf Publishing Company(Publisher)
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 andINDEPENDENCE 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 ifIt 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 obtainsThus, 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, thenIf 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. ThenThis 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:Acand B , A andBc,AcandBc.It is sufficient to prove these properties just for any one of these pairs. On choosing, for example, A andBc, one obtainsABc= A − B = A − AB
eBook - ePub- Perumal Mariappan(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
8 Probability 8.1 IntroductionThe 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. Clearly the occurrence of head prevents the occurrence of the tail, which implies that the 2 events are said to be mutually exclusive.
eBook - ePub- Roy Weatherford(Author)
- 2022(Publication Date)
- Routledge(Publisher)
103 This may well be so. 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.104Probability, 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).105The 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)
D) as follows:3.6 The concept of independenceIntuitively, 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 thatIn 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 A1 , A2 , . . . , An are independent ifExample1. Returning to the example in Section 3.4, we can ask whether the event “window open” is independent of the event “door open”. We haveButThus, 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
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
A.• Partition: A collection of events A1 , A2 , …, Ak 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 which1. The total number of possible simple events, N, is finite, and2. Each simple event is equally likely.Then the probability of any event A is defined asPr ( 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. So, while the number of heads in ten tosses may not be exactly equal to five, the proportion of heads in n tosses of a fair coin will, on the whole, keep getting closer and closer to 1/2 as n keeps increasing indefinitely. In this sense the term probability is referring to the long run relative frequency of the event of getting a head, and the classical definition essentially states that this long run relative frequency can be eventually made as close to the ratio given in Equation (5.3)
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)
Figure 2.2 .Figure 2.2Mutually exclusive eventsThe 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) isand for multiple events, the probability of observing either event A, or event B, or event C, or event D, etc., isP a + P bIn 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.P a + P b + P c + P d e t cExercise 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. The picture is summarised in the following table with the assumption that there is a 10% chance of a false result (be it positive or negative).
