Classification of Probability.pptx
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The document defines probability as a measure of how likely an event is to occur, expressed as a ratio from 0 to 1. It discusses the basics of probability formulas and the three main types: theoretical, experimental, and axiomatic probability. Theoretical probability uses the ratio of favorable to total outcomes without experiments. Experimental probability is based on conducting actual experiments and recording results. Axiomatic probability sets axioms that apply to all probability types, including Kolmogorov's three axioms that define probability as a positive real number between 0 and 1.
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Probability is a mathematical measure of how likely events are to occur. It can be expressed as a fraction, decimal, or percentage between 0 and 1. A probability experiment involves possible outcomes that make up a sample space. An event is a subset of outcomes. Theoretical probability calculates the likelihood of an event based on equally likely outcomes. Experimental probability is based on observed frequencies. Subjective probability relies on estimates rather than calculations. As experiments are repeated, experimental probability approaches theoretical probability due to the law of large numbers.
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CLASSICAL PROBABILITY.pptx
Classical probability is an approach based on the assumptions that random processes have a set of possible outcomes and each outcome is equally likely. There are three main types of probabilities: theoretical, experimental, and axiomatic. Theoretical probability is the ratio of favorable to possible outcomes. Experimental probability is the frequency of an outcome from experiments. Axiomatic probability describes probability through predefined axioms to ease calculating event occurrences.
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Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
probability-181112173236.pdf
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
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This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
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This document discusses key concepts in probability including experiments, outcomes, sample spaces, classical probability, empirical probability, subjective probability, complementary events, and the law of large numbers. Probability can be calculated classically by considering the number of outcomes in an event over the total number of outcomes, empirically by observing frequencies, or subjectively based on estimates. Understanding probability is important for properly evaluating risks and uncertainties.
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This document discusses key concepts and terms related to probability. It begins by defining probability and explaining how it is used in various fields like science, commerce, and weather forecasting. Some key terms are then defined, such as outcome, event, experiment, trial, elementary event, and sample space. The document outlines two types of probability - experimental probability, which is based on empirical results from repeated experiments, and theoretical probability, which assumes all outcomes are equally likely. It provides examples of common experiments like coin tosses, die rolls, and card draws and lists their possible outcomes. Finally, it discusses the range of probability from 0 to 1 and types of events like sure, impossible, and complementary events.
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Probability is a mathematical measure of how likely events are to occur. It can be expressed as a fraction, decimal, or percentage between 0 and 1. A probability experiment involves possible outcomes that make up a sample space. An event is a subset of outcomes. Theoretical probability calculates the likelihood of an event based on equally likely outcomes. Experimental probability is based on observed frequencies. Subjective probability relies on estimates rather than calculations. As experiments are repeated, experimental probability approaches theoretical probability due to the law of large numbers.
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Probability is a measure of how likely outcomes of uncertain events are to occur. It can be determined through experiments repeated over the long-term to reveal consistent patterns (the classical method), by observing the proportion of outcomes in many trials of an experiment (the empirical method), or based on subjective judgments of likelihood (the subjective method). The probability of any given event is calculated as the number of desired outcomes divided by the total number of possible outcomes, assuming all outcomes are equally likely.
CLASSICAL PROBABILITY.pptx
Classical probability is an approach based on the assumptions that random processes have a set of possible outcomes and each outcome is equally likely. There are three main types of probabilities: theoretical, experimental, and axiomatic. Theoretical probability is the ratio of favorable to possible outcomes. Experimental probability is the frequency of an outcome from experiments. Axiomatic probability describes probability through predefined axioms to ease calculating event occurrences.
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Probability is a branch of mathematics that studies patterns of chance. It is used to quantify the likelihood of events occurring in experiments or other situations involving uncertainty. The probability of an event is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 indicating certainty. Key concepts in probability include theoretical and experimental probability, sample spaces, events, mutually exclusive and exhaustive events, and rules like addition rules for calculating combined probabilities. Probability is applied in many fields including statistics, gambling, science, and machine learning.
probability-181112173236.pdf
This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
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This document discusses key concepts in probability. It defines basic terms like experiment, sample space, event, and probability. It provides examples of calculating probability for coin tosses and dice rolls using the classical method of dividing the number of ways an event can occur by the total number of possible outcomes. The document also discusses limitations of the classical method and introduces the empirical and subjective methods of determining probability based on observed frequencies and personal judgment respectively.
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This document discusses key concepts in probability including experiments, outcomes, sample spaces, classical probability, empirical probability, subjective probability, complementary events, and the law of large numbers. Probability can be calculated classically by considering the number of outcomes in an event over the total number of outcomes, empirically by observing frequencies, or subjectively based on estimates. Understanding probability is important for properly evaluating risks and uncertainties.
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This document discusses key concepts and terms related to probability. It begins by defining probability and explaining how it is used in various fields like science, commerce, and weather forecasting. Some key terms are then defined, such as outcome, event, experiment, trial, elementary event, and sample space. The document outlines two types of probability - experimental probability, which is based on empirical results from repeated experiments, and theoretical probability, which assumes all outcomes are equally likely. It provides examples of common experiments like coin tosses, die rolls, and card draws and lists their possible outcomes. Finally, it discusses the range of probability from 0 to 1 and types of events like sure, impossible, and complementary events.
1616 probability-the foundation of probability theory
1) A probabilistic experiment is one where more than one outcome is possible and the outcome is uncertain. The sample space is the set of all possible outcomes.
2) Elementary events are the individual outcomes, while compound events are unions of elementary events. Probability axioms state that probabilities of events must be between 0 and 1, the probability of the sample space is 1, and the probabilities of disjoint events sum to the probability of their union.
3) There are three fundamental theorems of probability: the probability of the empty set is 0; the probability of the entire sample space is 1; and the probability of the union of two events equals the sum of their probabilities minus the probability of their intersection.
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Probability is a numerical measure of how likely an event is to occur on a scale of 0 to 1. It can be used to quantify uncertainty. The classical definition of probability assigns an equal likelihood to all outcomes of an experiment. The relative frequency definition is based on the number of times an event occurs over many trials of an experiment. Subjective probability relies on intuition to assign likelihoods. Events can be simple, consisting of a single outcome, or compound, consisting of multiple outcomes. Mutually exclusive events cannot both occur, so their joint probability is zero.
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Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.[note 1][1][2] The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable; the probability of 'heads' equals the probability of 'tails'; and since no other outcomes are possible, the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
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2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3
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Probability is a branch of mathematics that studies patterns of chance and uncertainty. It is expressed as a number between 0 and 1 that indicates how likely an event is to occur based on the number of possible outcomes. For example, the probability of flipping a coin and getting heads is 1/2 or 50% since there are two possible outcomes and one favors heads. Probability can be theoretical, based on the number of outcomes, or experimental, based on recording results of repeated trials. Key terms include experiment, outcome, sample space, event, and probability definitions for single and multiple events.
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Probability is a measure of how likely an event is to occur. It is expressed as a fraction between 0 and 1, where the numerator is the number of favorable outcomes and the denominator is the total number of possible outcomes. There are general rules for probability - the probabilities of all outcomes must sum to 1, and the probability of an event not occurring is 1 minus the probability of it occurring. Probability is used in many fields involving chance, like math, statistics, finance, science, and more.
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1616 probability-the foundation of probability theory
1) A probabilistic experiment is one where more than one outcome is possible and the outcome is uncertain. The sample space is the set of all possible outcomes.
2) Elementary events are the individual outcomes, while compound events are unions of elementary events. Probability axioms state that probabilities of events must be between 0 and 1, the probability of the sample space is 1, and the probabilities of disjoint events sum to the probability of their union.
3) There are three fundamental theorems of probability: the probability of the empty set is 0; the probability of the entire sample space is 1; and the probability of the union of two events equals the sum of their probabilities minus the probability of their intersection.
What is probability
Probability is a numerical measure of how likely an event is to occur on a scale of 0 to 1. It can be used to quantify uncertainty. The classical definition of probability assigns an equal likelihood to all outcomes of an experiment. The relative frequency definition is based on the number of times an event occurs over many trials of an experiment. Subjective probability relies on intuition to assign likelihoods. Events can be simple, consisting of a single outcome, or compound, consisting of multiple outcomes. Mutually exclusive events cannot both occur, so their joint probability is zero.
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Probability is the branch of mathematics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur.[note 1][1][2] The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is fair, the two outcomes ('heads' and 'tails') are both equally probable; the probability of 'heads' equals the probability of 'tails'; and since no other outcomes are possible, the probability of either 'heads' or 'tails' is 1/2 (which could also be written as 0.5 or 50%).
These concepts have been given an axiomatic mathematical formalization in probability theory, which is used widely in areas of study such as statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, game theory, and philosophy to, for example, draw inferences about the expected frequency of events. Probability theory is also used to describe the underlying mechanics and regularities of complex systems.
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The document discusses the history and concepts of probability. It outlines that probability theory originated from a gambling dispute in the 17th century and developed rapidly in the 18th century. It defines probability as a number between 0 and 1 that reflects the likelihood of an event occurring. It also classifies the different types of probabilities as theoretical, experimental, and axiomatic probability and provides examples of where probability is used in real life situations and how it can help develop critical thinking skills.
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This document defines key concepts in probability and provides examples. It discusses probability vocabulary like sample space, outcome, trial, and event. It defines probability as the number of times a desired outcome occurs over total trials. Events are independent if the outcome of one does not impact others, and mutually exclusive if they cannot occur together. The addition and multiplication rules for probability are explained. Conditional probability describes the probability of a second event depending on the first occurring. Counting techniques are discussed for finding total possible outcomes of combined experiments. Review questions are provided to test understanding of the material.
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- Key terms like sample space, trial, random experiment, and classical definition of probability.
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Probability is a branch of mathematics used to quantify the likelihood of events occurring based on patterns of chance observed in experiments. It is defined as the ratio of favorable outcomes to total possible outcomes. Common probability terms include experiment, outcome, equally likely outcomes, sample space, event, and sample point. Probability calculations follow set rules and can be applied in various domains like gambling, science, and artificial intelligence.
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1) 1.56 and 1.0 cannot be probabilities because probabilities must be between 0 and 1.
2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3
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This document provides an overview of probability theory concepts. It defines an experiment as any process of observation or measurement, and a random experiment as one where the exact outcome cannot be predicted but the possible outcomes can be listed. The sample space is the set of all possible outcomes, and a subset is called an event. Probability is defined as the ratio of favorable outcomes to total possible outcomes. Examples are provided of calculating probabilities of events occurring for experiments like rolling dice, tossing coins, and assigning student grades.
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This document discusses key concepts in probability theory including:
- Probability is a quantitative measure of uncertainty ranging from 0 to 1.
- Experiments produce outcomes that define events in a sample space.
- There are different approaches to determining probability (classical, relative frequency, subjective).
- Probability can be classified as marginal, union, joint, or conditional depending on the relationship between events.
- Common discrete and continuous probability distributions include the binomial, Poisson, and normal distributions.
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This document provides an introduction to probability. It defines probability as a measure of how likely an event is to occur. Probability is expressed as a ratio of favorable outcomes to total possible outcomes. The key terms used in probability are defined, including event, outcome, sample space, and elementary events. The theoretical approach to probability is discussed, where probability is predicted without performing the experiment. Random experiments are described as those that may not produce the same outcome each time. Laws of probability are presented, such as a probability being between 0 and 1. Applications of probability in everyday life are mentioned, such as reliability testing of products. Two example probability problems are worked out.
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Presentation (2).pptx
Probability is a branch of mathematics that studies patterns of chance and uncertainty. It is expressed as a number between 0 and 1 that indicates how likely an event is to occur based on the number of possible outcomes. For example, the probability of flipping a coin and getting heads is 1/2 or 50% since there are two possible outcomes and one favors heads. Probability can be theoretical, based on the number of outcomes, or experimental, based on recording results of repeated trials. Key terms include experiment, outcome, sample space, event, and probability definitions for single and multiple events.
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This document provides an overview of key concepts in probability. It defines probability as a measure of likelihood between 0 and 1, and discusses how probability is used in various fields. Key terms are explained, such as randomness, trial, independent trial, experiment, event, and sample space. Randomness refers to experiments with unpredictable outcomes, like rolling a die. Trials are the individual occurrences of an experiment, like each coin toss. The document also discusses empirical probability, which is the probability of an event calculated based on the frequency of outcomes from experiments. Empirical probability depends on the specific experiment and can take on different values.
Probability
Probability is a measure of how likely an event is to occur. It is expressed as a fraction between 0 and 1, where the numerator is the number of favorable outcomes and the denominator is the total number of possible outcomes. There are general rules for probability - the probabilities of all outcomes must sum to 1, and the probability of an event not occurring is 1 minus the probability of it occurring. Probability is used in many fields involving chance, like math, statistics, finance, science, and more.
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Prokaryotic DNA replication begins at the origin of replication and proceeds bidirectionally. It involves three main steps: initiation, elongation, and termination. Initiation begins at the origin of replication site, where proteins bind and unwind the DNA. Elongation then uses DNA polymerase III to simultaneously synthesize the leading and lagging strands. Termination occurs when the replication forks from opposite directions meet at the termination site and halt, resolving issues through proteins like TUS and topoisomerases.
Prokaryotic DNA replication.pptx
Prokaryotic DNA replication begins at the origin of replication site where the replication fork is formed. DnaA and other proteins recognize the origin site and unwind the DNA double helix. RNA primers are synthesized which allows DNA polymerase III to begin bidirectional DNA synthesis of the leading and lagging strands. Okazaki fragments are formed and ligated on the lagging strand. Replication terminates when the replication forks meet at the termination site and the daughter chromosomes are separated.
DNA repair mechanism and their types.pptx
DNA repair is a collection of processes cells use to identify and correct damage to DNA. There are several types of DNA repair mechanisms, including direct repair, base excision repair, mismatch repair, recombination repair, and SOS repair. DNA repair has advantages of maintaining genetic integrity between generations and preserving individual health, but defects can lead to increased cancer risks and premature aging syndromes.
TOPIC NAME Structure of DNA and their Type.pptx
DNA has a double helix structure that resembles a twisted ladder. It is composed of nucleotides, each containing a sugar, phosphate group, and one of four nitrogen bases (A, T, C, or G). The bases on opposite strands bond together via hydrogen bonds between A-T and C-G to form the structure. Erwin Chargaff discovered that the amounts of A and T are equal, as are the amounts of C and G in any DNA sample. There are three types of DNA structures: A-DNA, B-DNA, and Z-DNA.
LAC OPERON SYSTEM.pptx
The document describes the lac operon system, which was the first operon described in 1961 and controls the catabolism of lactose in prokaryotes. The lac operon has 5 structural units - a promoter, operator, three structural genes (LacZ, LacY, LacA) that encode enzymes for lactose breakdown, CAP binding sites, and a regulatory gene. When lactose is absent, a repressor protein binds to the operator, preventing transcription of the structural genes. When lactose is present, it binds to the repressor and causes a conformational change that releases it from the operator, allowing transcription and the breakdown of lactose.
LAC OPERON 2.pptx
The lac operon is a cluster of three genes in prokaryotes that are involved in processing the sugar lactose. It consists of a promoter, operator, and structural genes. The lac operon is inducible, meaning it is only transcribed when lactose is present. Depending on whether lactose and glucose are present or absent, the lac operon will be either actively transcribed or transcriptionally inactive.
DNA STRUCTURE AND TYPES.pptx
DNA is a double-stranded molecule composed of nucleotides that contain nitrogenous bases, a deoxyribose sugar, and a phosphate group. The bases--cytosine, thymine, adenine, and guanine--form rungs of a twisted ladder structure, while the alternating sugar and phosphate molecules form the sides. Watson and Crick discovered that the bases pair up specifically through hydrogen bonds to form the DNA double helix, which contains the genetic information needed to build an organism. DNA can take on different secondary structures including A-DNA, B-DNA, and Z-DNA forms.
TOPIC-DNA repair mechanism and their types.pptx
DNA repair mechanisms are necessary to correct damage to DNA that could otherwise lead to mutations and cancer. There are several types of DNA repair mechanisms that address different types of damage, including base repair through excision, resynthesis and ligation of bases, nucleotide excision repair to address mismatches, and mismatch repair to fix errors. Double stranded breaks can be repaired through non-homologous end joining or homologous recombination repair.
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Classification of Probability.pptx
- 2. Probability is the measure of how likely it is some event will occur, a number expressing the ratio of favourable cases to the whole number or cases possible. The value is expressed from zero to one. The idea of probability is based on observation.
- 3. BASICS FORMULA OF PROBABILITY Probability of event to happen P(E) = Number of favourable outcomes/Total Number of outcomes [Where E is event.]
- 4. TYPES OF PROBABILITY Theoratical Probability Experimental Probability Axiomatic Probability
- 5. THEORETICAL PROBABILITY The theoretical probability is defined as the ratio of the number of favourable outcomes to the number of possible outcomes. FORMULA OF THEORETICAL PROBABILITY Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes.
- 6. Example of Theoratical Probability Find the probability of rolling a 5 on a fair die Solution: To find the probability of getting 5 while rolling a die, an experiment is not needed. We know that there are 6 possible outcomes when rolling a die. They are 1, 2, 3, 4, 5, 6. Therefore, the probability is, Probability of Event P(E) = No. of. Favourable outcomes/ No. of. Possible outcomes. P(E) = 1/6. Hence, the probability of getting 5 while rolling a fair die is 1/6.
- 7. EXPERIMENTAL PROBABILITY Experimental probability, also known as Empirical probability, is based on actual experiments and adequate recordings of the happening of events. To determine the occurrence of any event, a series of actual experiments are conducted. FORMULA OF EXPERIMENTAL PROBABILITY Probability of an Event P(E) = Number of times an event occurs / Total number of trials.
- 8. Example of Experimental Probability
- 9. AXIOMATIC PROBABILITY Axiomatic probability is a unifying probability theory. It sets down a set of axioms (rules) that apply to all of types of probability, including frequentist probability and classical probability. These rules, based on Kolmogorov’s Three Axioms, set starting points for mathematical probability.
- 10. THESE THREE AXIOMS ARE First Axiom The probability of an event is a positive real number, P(E)≥0 Second Axiom The probability of the sum of all subsets in the sample space is P(S) = 1(OR)P(ω1)+P(ω2)+…P(ωn)=1 Third Axiom If E1 and E2 are mutually exclusive events, thenP(E1∪E2)=P(E1)+P(E2) We can also see this true for P(ϕ). Therefore, P(E∪ϕ)=P(E)+P(ϕ)=P(E)Here, P(ϕ) is a null set (or) P(ϕ) = 0
- 11. APPLICATIONS OF PROBABILITY IN REAL LIFE Weather planning Sports strategies Insurance In Games In Politics