Probability’s definition

Probability
is the chance that something will occur though it is that events will occur.
Often you can calculate a probability with a number like “15%
chance”, or you can use expressions like impossible, unlikely, and
possible, even chance, likely and certain.

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Mathematics
has many branches one of them is probability which is expressed as a number
between 0 and 1, and that’s measured by that kind given by the occurrence of real
occurrence.

For
example the probability of coin toss has only two options either ”tails” or
”heads” this case is considered a probability of one.

Probability
of 0.5 is believed to include similar odds if happening or not happening such
as the probability of a coin toss resulting ”heads” or ”tails” but for the
probability of zero is believed to be impossibility, in this case the coin will
land flat without either side facing up that is zero that’s why ”head” or
”tails” must be facing up

It’s
the easiest way can be mathematically
considered as the number of occurrence of specific event divided by the
number of occurrence added to the number of failures of occurrence ( this adds
up to the total of possible outcomes)  Pa=Pa/ (Pa+Pb)

When
a single die is thrown , there’s six possible outcomes :1 , 2 , 3 , 4 , 5 , 6

The
probability of one of them is 1/6

Probability
theorems:

Bapat-beg
theorem : In probability theory, the Bapat–Beg theorem provides the joint
probability distribution of order statistics of independent

All components of the sample are gained from the same population and
thus have the same probability
distribution, and the
Bapat–Beg theorem shows the order statistics when each component of the sample
is gained from a various statistical population and therefore has its own probability
distribution.

Markov-krein theorem:

It states that the predicted values of real function of random variables
where only the early moments of random variable are known.

Craps principle theorem:

it’s the theory which talks about event
probabilities below Independent
and identically distributed random variables
trails , as E1 and E2 gives two mutually exclusive events which may happen on a
given trial.

Types of random variables:

A
Random Variable is a set of considerable
significances from a random experiment.

There are two types of random variables:

1-Discrete random variable:

It has limited available significances or
an unlimited series of certain numbers

– X:  number of hits on trying 40 free throws.

2-
Continuous random variable:

It
takes all uncountable values in a period of real numbers

– X:  the period it
takes for a lamp to burn.

Types
of probability distributions:

1-Geometric
distribution:

On independent Bernoulli trials are
made over and over, and each with probability (p) of success, the number of
trials (X), and it takes to get the first success has a geometric distribution.

2-Negative
binomial distribution:

Each with probability (P) of
success, and (X) is the trial number when (r) successes are first accomplished,
then X has a non-positive binomial distribution. PS: that Geometric (p) =
Negative Binomial (p, 1).

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