Probability’s definition Probabilityis 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. Mathematicshas many branches one of them is probability which is expressed as a numberbetween 0 and 1, and that’s calculated by that branch given by the occurrenceof certain event.
Forexample the probability of coin toss has only two options either ”tails” or”heads” this case is considered a probability of one.Probabilityof 0.5 is believed to include same odds if happening or not happening such asthe probability of a coin toss resulting ”heads” or ”tails” but for theprobability of zero is believed to be impossibility, in this case the coin willland flat without either side facing up that is zero that’s why ”head” or”tails” must be facing up It’sthe easiest way can be mathematically considered as the number of occurrence of specific event divided by thenumber of occurrence added to the number of failures of occurrence ( this addsup to the total of possible outcomes) Pa=Pa/ (Pa+Pb)Whena single die is thrown , there’s six possible outcomes :1 , 2 , 3 , 4 , 5 , 6Theprobability of one of them is 1/6Probabilitytheorems:Bapat-begtheorem : In probability theory, the Bapat–Beg theorem provides the jointprobability distribution of order statistics of independent All components of the sample are gained from the same population andthus have the same probabilitydistribution, and theBapat–Beg theorem shows the order statistics when each component of the sampleis gained from a various statistical population and therefore has its own probabilitydistribution.Markov-krein theorem:It states that the predicted values of real function of random variableswhere only the early moments of random variable are known.Craps principle theorem: it’s the theory which talks about eventprobabilities below Independentand identically distributed random variablestrails , as E1 and E2 gives two mutually exclusive events which may happen on agiven trial. Types of random variables: ARandom Variable is a set of considerablesignificances from a random experiment.
There are two types of random variables:1-Discrete random variable:It has limited available significances oran unlimited series of certain numbers – X: number of hits on trying 40 free throws. 2-Continuous random variable:Ittakes all uncountable values in a period of real numbers– X: the period ittakes for a lamp to burn.Typesof probability distributions:1-Geometricdistribution:On independent Bernoulli trials aremade over and over, and each with probability (p) of success, the number oftrials (X), and it takes to get the first success has a geometric distribution.2-Negativebinomial distribution:Each with probability (P) ofsuccess, 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).