Probability’s when each component of the sample is

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 Probability’s definition

is the chance that something will happen however it is that events will occur.
Sometimes you can measure a probability with a number like “10%
chance”, or you can use words such as impossible, unlikely, and possible,
even chance, likely and certain. 

has many branches one of them is probability which is expressed as a number
between 0 and 1, and that’s calculated by that branch given by the occurrence
of certain event.

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

of 0.5 is believed to include same 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

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)

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

probability of one of them is 1/6


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

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:

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.






Continuous random variable:

takes all uncountable values in a period of real numbers

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

of probability distributions:


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

binomial distribution:

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

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