Let`s use this addition rule to determine the probability for experiment 1. This is reflected in the following formulas: The formula for calculating the probability of two events A and B is given by: Solution: Be F = the event that the student verifies the fiction; and leave N = the event where the student consults non-fiction books. Then, based on the addition rule: The first formula is only the sum of the probabilities of the two events. The second formula is the sum of the probabilities of the two events minus the probability that both will occur. Bayesian inference is an inference method that uses Bayes` rule to update the estimate of the probability of a hypothesis as additional evidence is learned. Bayesian updating is an important technique in all statistics and especially in mathematical statistics. Bayesian updating is particularly important in the dynamic analysis of a data sequence. Bayesian inference has found application in a number of fields, including science, engineering, philosophy, medicine, and law. The distribution of probability of addition (sometimes called the addition rule or sum rule) states that the probability that [latex]text{A}[/latex] or [latex]text{B}[/latex] occurs is the sum of probabilities that [latex]text{A}[/latex] will pass and that [latex]text{B}[/latex] will occur, minus the probability that [latex]text{A}[/latex] and [latex]text{B}[/latex] will occur. The rule of addition is summarized by the formula: Experience 3: A glass contains 1 red ball, 3 green, 2 blue and 4 yellow. If only one ball is chosen at random from the glass, what is the probability that it is yellow or green? Let`s take the following example. If you draw a card from a stack of playing cards [latex]52[/latex], what is the probability of getting a heart or an image card (king, queen or jack)? Let [latex]text{H}[/latex] be the drawing of a core and [latex]text{F}[/latex] the drawing of an illustrated map. Since there are hearts [latex]13[/latex] and a total of face cards [latex]12[/latex] ([latex]3[/latex] of any color: spades, hearts, diamonds and clubs), but only face cards [latex]3[/latex] of hearts, we get: Suppose a card is drawn from a deck of 52 playing cards: What is the probability of getting a king or a lady? Be [latex]text{A}[/latex] for the event where a king is drawn and [latex]text{B}[/latex] for the event where a queen is drawn.

These two events are disjointed because there are no kings who are also queens. So, in reality, both rules are simplified to a single rule, the second. Indeed, in the first case, the probability of two mutually exclusive events occurring is 0. In the example with the cube, it is impossible to roll both a 3 and a 6 on a coil of a single cube. The two events are therefore mutually exclusive. Consider a good roll of the dice, which is another example of independent events. If one person rolls two, the result of the first role does not change the probability of the outcome of the second role. Multiplication rule The probability that events A and B will occur is equal to the probability that event A will occur multiplied by the probability that event B will occur, provided that A has occurred. To illustrate the second rule, consider a class in which there are 9 boys and 11 girls. At the end of the semester, 5 girls and 4 boys receive a grade of B.

If a student is chosen at random, what is the probability that he or she is a girl or a student B? Since the chances of choosing a girl are 11 to 20, the chances of choosing a B student are 9 out of 20, and the chances of choosing a girl who is a B student are 5/20, the chances of choosing a girl or a B student are: Before discussing the rules of probability, let`s give the following definitions: ExampleA student goes to the library. The probability that she (a) will consult a work of fiction is 0.40, (b) a non-fiction book is 0.30, and (c) fiction and non-fiction are 0.20. How likely is it that the student will consult a fiction book, a non-fiction book, or both? The People of the State of California v. Collins was a jury trial in 1968 in California. He made notorious medico-legal use of statistics and probabilities. Passers-by on a robbery in Los Angeles testified that the perpetrators were a black man with a beard and mustache and a Caucasian woman with blond hair tied to a ponytail. They had escaped in a yellow car. We obtain the general rule of multiplication by multiplying both sides of the conditional probability definition by the denominator. That is, in the equation [latex]displaystyle text{P}(text{A}|text{B})=frac{text{P}(text{A}cap text{B})}{text{P}(text{B})}[/latex], if we multiply both sides by [latex]text{P}(text{B})[/latex], we get the multiplication rule. Bayes` rule expresses how a subjective degree of belief should change rationally to accommodate evidence. Experiment 2: A spinner has 4 equal sectors colored in yellow, blue, green and red.

What is the probability of landing on red or blue after turning this spinner? The addition rule for probabilities describes two formulas, one for probability for one of the two mutually exclusive events, and the other for the probability that two non-mutually exclusive events occur. A card is drawn at random from a game of ordinary playing cards. You earn \$10 if the card is a spade or ace. How likely are you to win the game? If events A and B are not independent of each other, the probability may be derived from the nature of the events, or it is otherwise difficult to determine. In a previous lesson, we learned two important properties of probability: First, note that each coin throw is an independent event. The side on which a piece lands does not depend on what has happened before. In probability theory and statistics, Bayes` theorem (or Bayes` rule) is an important result in the mathematical manipulation of conditional probabilities. This is a result that results from the more elementary axioms of probability. When applied, the probabilities involved in Bayes` theorem can have one of many probability interpretations. In one of these interpretations, the theorem is used directly as part of a particular approach to statistical inference. In particular, with the Bayesian interpretation of probability, the theorem expresses how a subjective degree of belief should change rationally to take into account the proofs.

This is called Bayesian inference, which is fundamental to Bayesian statistics. In the case of mutually exclusive eventsIn statistics and probability theory, two events are mutually exclusive if they cannot occur at the same time. The simplest example of mutual exclusion is that the probability that both events occur at the same time is by definition zero, because if one occurs, the other event cannot do so. For mutually exclusive events A and B, then there is: The Collins case is an excellent example of a phenomenon known as prosecutor`s error – an error in statistical reasoning when used as an argument in court proceedings. Basically, error implies the assumption that the previous probability of a random match is equal to the probability that the defendant is innocent. For example, if it is known that an aggressor has the same blood type as an accused (and that 10% of the population shares that blood type), it is wrong for the charge (in a very simple form) to argue on this basis alone that the probability that the accused will be guilty is 90%. .