Question

The following corepts mathematically and ve some examples (numerically which are necessary in - probability theory event Somple spoke statistical inference descriptive s -permutasyon kombinasyon - density function conditional probability - multiplication rule -dependent event - Independent event Bojes! Rule. P.S! The find Bayes' rule. and discuit the validated of 1

Answer 1

Probability theory
Is the branch of mathematics that deals with quantities having random distributions. It is concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance.

Event
In probability theory, an event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned. If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one).

Sample Space
In probability theory, the sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).

Statistical inference
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution.Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.

Descriptive statistic
A descriptive statistic is a summary statistic that quantitatively describes or summarizes features of a collection of information. Some measures that are commonly used to describe a data set are measures of central tendency and measures of variability or dispersion. Measures of central tendency include the mean, median and mode, while measures of variability include the standard deviation (or variance), the minimum and maximum values of the variables, kurtosis and skewness.

Prrmutation
In mathematics, permutation is the act of arranging the members of a set into a sequence or order, or, if the set is already ordered, rearranging (reordering) its elements—a process called permuting. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n.

Combination
n mathematics, a combination is a selection of items from a collection, such that (unlike permutations) the order of selection does not matter. For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange.

PDF
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space.

Conditional Probability
In probability theory, conditional probability is a measure of the probability of an event occurring given that another event has (by assumption, presumption, assertion or evidence) occurred.[1] If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A | B), or sometimes PB(A) or P(A / B). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability that someone coughing is unwell might be 75%, then: P(Cough) = 5%; P(Sick | Cough) = 75%

Multiplication Rule
The Multiplication Rule of Probability means to find the probability of the intersection of two events, multiply the two probabilities.
If A and B are two independent events in a probability experiment, then the probability that both events occur simultaneously is:
P(A and B)=P(A)⋅P(B)
In case of dependent events , the probability that both events occur simultaneously is:
P(A and B)=P(A)⋅P(B | A)

Dependent events
Independent events are when the probability of an event is not affected by a previous event.

Independent events
A dependent event is when one event influences the outcome of another event in a probability scenario.
To find the intersection of two events, whether they are independent or dependent, multiply the two probabilities together.

Bayes Rule
In probability theory and statistics, Bayes’ theorem (alternatively Bayes’ law or Bayes’ rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example, if cancer is related to age, then, using Bayes’ theorem, a person's age can be used to more accurately assess the probability that they have cancer than can be done without knowledge of the person’s age.
There are several different ways to write the formula for Bayes' theorem. The most common form is:
P(A ∣ B) = P(B ∣ A)P(A) / P(B)
where A and B are two events and P(B) ≠ 0
P(A ∣ B) is the conditional probability of event A occurring given that B is true.
P(B ∣ A) is the conditional probability of event B occurring given that A is true.
P(A) and P(B) are the probabilities of A and B occurring independently of one another (the marginal probability).