Examples of probability theory in the following topics:
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- Probability theory uses logic and mathematical reasoning, rather than experimental data, to determine probable outcomes.
- Mathematically, probability theory formulates incomplete knowledge pertaining to the likelihood of an event.
- As such, the meteorologist's 60% verdict is a theoretical probability, and not the result of any proven experiment.
- For example, the probability of rolling any specific number on a six-sided die is one out of six: there are six, equally probable sides to land on, and each side is distinct from the others.
- This is a theoretical probability; testing by rolling the die many times and recording the results would result in an experimental probability.
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- Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
- In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
- The probability for the random variable to fall within a particular region is given by the integral of this variable's probability density over the region.
- For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:
- Apply the ideas of integration to probability functions used in statistics
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- Probability is a mathematical tool used to study randomness.
- The expected theoretical probability of heads in any one toss is 1/2 or 0.5.
- The theory of probability began with the study of games of chance such as poker.
- Predictions take the form of probabilities.
- You might use probability to decide to buy a lottery ticket or not.
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- In probability theory, the probability $P$ of some event $E$, denoted $P(E)$, is usually defined in such a way that $P$ satisfies a number of axioms, or rules.
- Probability is a number.
- If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.
- The probability that an event does not occur is $1$ minus the probability that the event does occur.
- The probability that an event occurs and the probability that it does not occur always add up to $100\%$, or $1$.
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- Inferential statistics is built on the foundation of probability theory, and has been remarkably successful in guiding opinion about the conclusions to be drawn from data.
- Therefore the probability of heads is taken to be 1/2, as is the probability of tails.
- Of course, wind direction also affects probability.
- Questions such as "What is the probability that Ms.
- An event with probability 0 has no chance of occurring; an event of probability 1 is certain to occur.
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- The conditional probability of an event is the probability that an event will occur given that another event has occurred.
- Each individual outcome has probability $1/8$.
- Then the probability of $B$ given $A$ is $1/2$, since $A \cap B=\{HHH\}$ which has probability $1/8$ and $A=\{HHH,TTT\}$ which has probability $2/8$, and $\frac{1/8}{2/8}=\frac{1}{2}.$
- The conditional probability $P(B|A)$ is not always equal to the unconditional probability $P(B)$.
- In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule) is a result that is of importance in the mathematical manipulation of conditional probabilities.
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- If a theory can accommodate all possible results then it is not a scientific theory.
- Although strictly speaking, disconfirming an hypothesis deduced from a theory disconfirms the theory, it rarely leads to the abandonment of the theory.
- Instead, the theory will probably be modified to accommodate the inconsistent finding.
- This can lead to discontent with the theory and the search for a new theory.
- If a new theory is developed that can explain the same facts in a more parsimonious way, then the new theory will eventually supersede the old theory.
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- It is a probability calculated from experience, not from theory.
- Experimental probability contrasts theoretical probability, which is what we would expect to happen.
- In statistical terms, the empirical probability is an estimate of a probability.
- An advantage of estimating probabilities using empirical probabilities is that this procedure includes few assumptions.
- A disadvantage in using empirical probabilities is that without theory to "make sense" of them, it's easy to draw incorrect conclusions.
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- A continuous probability distribution is a probability distribution that has a probability density function.
- Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero.
- In theory, a probability density function is a function that describes the relative likelihood for a random variable to take on a given value.
- Unlike a probability, a probability density function can take on values greater than one.
- The standard normal distribution has probability density function:
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- Sociological theory is developed at multiple levels, ranging from grand theory to highly contextualized and specific micro-range theories.
- Putnam's theory proposes:
- This element of Putnam's theory clearly illustrates the basic purpose of sociological theory.
- In short, Putnam's theory clearly encapsulates the key ideas of a sociological theory.
- In fact, it is probably more useful and informative to view theories as complementary.