#### Theory of Multinomial, Poisson and HyperGeometric

Multinomial Probabilities:

The extended binomial probability is termed as Multinomial Probability. The difference is that in multinomial experiment, there are more than two possible outcomes. Though, there are still a fixed number of independent trials, the probability of each and every outcome should stay constant from trial to trial.

Rather than using a combination, as in the case of binomial probability, the number of ways the outcomes can take place is done by using distinguishable permutations.

The illustration is shown below which is very helpful in comparison with the formula.

The probability that a person will pass a College Algebra class is 0.55, the probability which a person will withdraw prior to the class is completed is 0.40, and the probability which a person will fail the class is 0.05. Determine the probability that in a class of 30 students, accurately 16 pass, 12 withdraw and 2 fail.

Outcome         x      p(outcome)
Pass              16     0.55
Withdraw      12     0.40
Fail                2       0.05
Total             30      1.00

The probability can be determined by using this formula:

P = [30!/((16!) (12!) (2!)] * 0.55^16 * 0.40^12 * 0.05^2

Poisson Probabilities:

The French mathematician Simeon Poisson introduced Poisson Probabilities and it named after them. Poisson probabilities are helpful when there are a large number of independent trials with small probability of success on a single trial and the variables take place over a period of time. This can also be employed when a density of items is distributed over a particular area or volume.

P(x; λ) = (e-λ λx)/x!, x = 0, 1, 2…

The Lambda in formula is the mean number of occurrences. Whenever you are approximating a binomial probability by using the Poisson, then lambda is similar as mu or n * p.

Illustration: When there are 500 customers per 8-hour day in a check-out lane, then what is the probability that there will be accurately 3 in line throughout any five-minute period?

The predicted value throughout any one five minute period would be 500/96 = 5.2083333. The 96 is used as there are 96 five-minute periods in 8 hours. Therefore, you expect regarding 5.2 customers in 5 minutes and want to know the probability of getting accurately 3.

p(3;500/96) = e^(-500/96) * (500/96)^3 / 3! = 0.1288

Hypergeometric Probabilities:

Hypergeometric experiments take place whenever the trials are not independent of one other and take place due to sampling with no replacement -- as in a five card poker hand.

Hypergeometric probabilities include the multiplication of two combinations altogether and then divide by the total number of combinations.

Illustration: In how many ways can 3 men and 4 women be chosen from a group of 7 men and 10 women?

The answer is = 7350/19448 = 0.3779

It must be noted that the sum of numbers in numerator are the numbers utilized in the combination in denominator.

This can be expanded to more than two groups and are termed as extended hypergeometric problem.

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