Probability of Joint Occurrences

Let us study about
Probability of Joint Occurrences :

Another way to compute the probability of all three flipped coins landing heads is as a series of three different events: first flip the penny, then flip the nickel, then flip the dime. Will the probability of landing three heads still be .125?

Multiplication rule :

To compute the probability of two or more independent events all occurring— joint occurrence—multiply their probabilities.

For example, the probability of the penny landing heads is 1/2, or .5; the probability of the nickel next landing heads is 1/2, or .5; and the probability of the dime landing heads is 1/2, or .5; thus, note that

.5 x .5 x .5 = .125

which is what you determined with the classic theory by assessing the ratio of number of favorable outcomes to number of total outcomes. The notation for joint occurrence is P(AB) = P(A) × P(B) and reads: The probability of A and B both happening is equal to the probability of A times the probability of B.

Using the multiplication rule, you can also determine the probability of drawing two aces in a row from a deck of cards. The only way to draw two aces in a row from a deck of cards is for both draws to be favorable. For the first draw, the probability of a favorable outcome is 4/52. But because the first draw is favorable, only 3 aces are left among 51 cards. So the probability of a favorable outcome on the second draw is 3/51. For both events to happen, you simply multiply those two probabilities together:
Note that these probabilities are not independent. If, however, you had decided to return the initial card drawn back to the deck before the second draw, then the probability of drawing an ace on each draw is 4/52, as these events are now independent. Drawing an ace twice in a row, with the odds being 4/52 both times, gives the following:

In either case, you use the multiplication rule because you are computing probability for favorable outcomes in all events.
Hope the above information was helpful.

Definitions of the Determinant

Definitions of the Determinant :

The determinant function can be defined by essentially two different methods. The advantage of the first definition—one which uses permutations—is that it provides an actual formula for det A, a fact of theoretical importance. The disadvantage is that, quite frankly, no one actually computes a determinant by this method.

Method 1 for defining the determinant. If n is a positive integer, then a permutation of the set S = {1, 2, …, n} is defined to be a bijective function—that is, a one-to-one correspondence—σ, from S to S. For example, let S = {1, 2, 3} and define a permutation σ of S as follows:
Since σ(1) = 3, σ(2) = 1, and σ(3) = 2, the permutation σ maps the elements 1, 2, 3 into 3, 1, 2. Intuitively, then, a permutation of the set S = {1, 2, …, n} provides a rearrangement of the numbers 1, 2, …, n. Another permutation, σ′, of the set S is defined as follows:
This permutation maps the elements 1, 2, 3 into 2, 1, 3, respectively. This result is written
Hope the above explanation was useful, not let me explain you about adjoint matrix.

Introduction of Analytic Function

Let us learn about Analytic Function,

A complex function is said to be analytic on a region R if it is complex differentiable at every point in R. The terms holomorphic function, differentiable function, and complex differentiable function are sometimes used interchangeably with "analytic function" (Krantz 1999, p. 16). Many mathematicians prefer the term "holomorphic function" (or "holomorphic map") to "analytic function" (Krantz 1999, p. 16), while "analytic" appears to be in widespread use among physicists, engineers, and in some older texts (e.g., Morse and Feshbach 1953, pp. 356-374; Knopp 1996, pp. 83-111; Whittaker and Watson 1990, p. 83).

If a complex function is analytic on a region R, it is infinitely differentiable in R. A complex function may fail to be analytic at one or more points through the presence of singularities, or along lines or line segments through the presence of branch cuts.

A complex function that is analytic at all finite points of the complex plane is said to be entire. A single-valued function that is analytic in all but possibly a discrete subset of its domain, and at those singularities goes to infinity like a polynomial (i.e., these exceptional points must be poles and not essential singularities), is called a meromorphic function.
Hope the above explanation helped you.

What is Polygon

Let us study about polygon,
A polygon is a flat closed figure made from straight sides. In order to be a polygon, a figure needs to have at least three sides, but it can have more than that. Here are some polygons:

- the triangle, with three sides
- the quadrilateral, with four sides
- the pentagon, with five sides


If the sides are different lengths, then the shape of a polygon will not always be the same. There are many different triangles, for example, that are very different-looking, because their three sides are different from triangle to triangle.

If however the sides of the polygon are all the same length, then the polygon is called regular.

A regular triangle has three equal sides. It is sometimes called an equilateral triangle. It could also be called equiangular, because its three angles are equal too.
A regular quadrilateral has four sides equal and four angles equal. It is also called a square.

But not all polygons are regular.