The modern mathematical theory of probability has its roots in attempts to analyze games of chance by Gerolamo Cardano in the sixteenth century, and by Pierre de Fermat and Blaise Pascal in the seventeenth century for example the " problem of points ". Christiaan Huygens published a book on the subject in [4] and in the 19th century, Pierre Laplace completed what is today considered the classic interpretation.

Initially, probability theory mainly considered discrete events, and its methods were mainly combinatorial. Eventually, analytical considerations compelled the incorporation of continuous variables into the theory. This culminated in modern probability theory, on foundations laid by Andrey Nikolaevich Kolmogorov. Kolmogorov combined the notion of sample space , introduced by Richard von Mises , and measure theory and presented his axiom system for probability theory in This became the mostly undisputed axiomatic basis for modern probability theory; but, alternatives exist, such as the adoption of finite rather than countable additivity by Bruno de Finetti.

Most introductions to probability theory treat discrete probability distributions and continuous probability distributions separately. The measure theory-based treatment of probability covers the discrete, continuous, a mix of the two, and more. Consider an experiment that can produce a number of outcomes.

The set of all outcomes is called the sample space of the experiment. The power set of the sample space or equivalently, the event space is formed by considering all different collections of possible results.

For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. These collections are called events. If the results that actually occur fall in a given event, that event is said to have occurred. To qualify as a probability distribution , the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events events that contain no common results, e.

This event encompasses the possibility of any number except five being rolled. When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable. A random variable is a function that assigns to each elementary event in the sample space a real number. This function is usually denoted by a capital letter. This does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". Discrete probability theory deals with events that occur in countable sample spaces.

Examples: Throwing dice , experiments with decks of cards , random walk , and tossing coins. Classical definition : Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability. The modern definition does not try to answer how probability mass functions are obtained; instead, it builds a theory that assumes their existence [ citation needed ].

Continuous probability theory deals with events that occur in a continuous sample space. Classical definition : The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox. That is, F x returns the probability that X will be less than or equal to x. Whereas the pdf exists only for continuous random variables, the cdf exists for all random variables including discrete random variables that take values in R.

### Probability Theory: Collection of Problems

Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two. Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space :.

The measure corresponding to a cdf is said to be induced by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theoretic approach free of fallacies. For example, to study Brownian motion , probability is defined on a space of functions.

When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes. What is the probability of getting two consecutive tails? Example 2: Consider another example where a pack contains 4 blue, 2 red and 3 black pens.

If a pen is drawn at random from the pack, replaced and the process repeated 2 more times, What is the probability of drawing 2 blue pens and 1 black pen? When two events occur, if the outcome of one event affects the outcome of the other, they are called dependent events. Example 1: A pack contains 4 blue, 2 red and 3 black pens. If 2 pens are drawn at random from the pack, NOT replaced and then another pen is drawn. What is the probability of drawing 2 blue pens and 1 black pen?

Example 2: What is the probability of drawing a king and a queen consecutively from a deck of 52 cards, without replacement. Conditional probability is calculating the probability of an event given that another event has already occured. Example: A single coin is tossed 5 times. What is the probability of getting at least one head? What is the probability of the occurrence of a number that is odd or less than 5 when a fair die is rolled. We need to find P A or B.

A box contains 4 chocobars and 4 ice creams.

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## Publications of Erdos in Probability Theory

Tom eats 3 of them, by randomly choosing. What is the probability of choosing 2 chocobars and 1 icecream? When two dice are rolled, find the probability of getting a greater number on the first die than the one on the second, given that the sum should equal 8. A bag contains blue and red balls. Two balls are drawn randomly without replacement. The probability of selecting a blue and then a red ball is 0.

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The probability of selecting a blue ball in the first draw is 0. What is the probability of drawing a red ball, given that the first ball drawn was blue? A die is rolled thrice. What is the probability that the sum of the rolls is atleast 5.

## Probability: the basics

Learn how to solve: — Simple and compound interest problems — Speed, distance and time problems — Ratio and proportion — List of Maths Formulas. At least one hat is correctly returned is compliment that no hat is returned correctly. Two cards are drawn at random from an ordinary deck of 52 card. Find the probability P that a Both are spade b One is a spade and one is heart.

Solution please. Lets assume probability of picking a red ball is X. The personal director of a company wishes to select applicant for advanced training without regard to sex. Will you conclude that the applicants have arrived in a random fashion? The probability of snow tomorrow is 0.

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And the probability that it will bi colder is 0. The probability that it will not snow and not bi colder is 0. What is probability that it will not snow if it is colder tomorrow? There are three boxes, one of which contains a prize. A contestant is given two chances, such that if he chooses the wrong box in the first round, that box is removed from the selection and he then chooses between the two remaining boxes. What is the probability that the contestant wins? Above answer can be explained as Prob. XYZ company wants to start a food outlet in pakistan.

Determine probability of starting the outlet in: a saddar b defence area of any city c clifton given that the outlet is started in karachi. The probability that a randomly chosen sales prospect will make a purchase is 0. What is the probability of the next three deliveries are females? So,the probabilitt of the next three deliveries are females is 0. Five hundred raffle tickets are sold at P25 each for 3 pieces of P4,, P and P1, After each price drawing, the winner is then returned to the collection of tickets.

What is the expected value if the person purchases four 4 tickets? There are three routes from a person,s home to her place of work. How many ways can she go from her home to her office? If she makes her various choices at random,what is the probability that she will take mornungside drive,park in lot A,use the south entrance and take elevator 1.

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As she starts her car one morning, she recalls parking lot A and B are closed for repair. Three people get on an elevator that stops at three floors. Full Name Comment goes here. Are you sure you want to Yes No. Raoul at Univ of Amsterdam Tags christian robert probability. No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Theory of Probability revisited 1. Swirles eds. Theory of Probability.

Theory of Probability was the outcome, as a theory of inductive inference founded on the principle of inverse probability. No measure theoretic basis, e. This is the principle of inverse probability, given by Bayes in Theory of Probability revisited Fundamental notions The Bayesian framework Prior selection 2 Still perceives a potential problem Theory of Probability revisited Fundamental notions The Bayesian framework Posterior distribution Operates conditional upon the observations Incorporates the requirement of the Likelihood Principle Theory of Probability revisited Estimation problems Noninformative prior distributions Noninformative distributions Theory of Probability revisited Estimation problems Noninformative prior distributions Very shaky from a mathematical point of view Theory of Probability revisited Estimation problems Sampling models Laplace succession rule Example of a predictive distribution For this reason the uniform assessment must be abandoned for ranges including the extreme values.

Theory of Probability revisited Estimation problems Sampling models Another contradiction For the multinomial model Mr n; p1 ,. Theory of Probability revisited Estimation problems Normal models and linear regression Laplace approximation ToP presents the normal distribution as a second order approximation of slowly varying densities, n P dx x1 ,.