Binomial Distribution

The binomial distribution is used when you have only two outcomes, and the outcomes are independent. So, for example, you can have conforming or non-conforming output in a process, and you want to count the number of non-conforming.


Example: I want to reply to the question, “If the probability of non-conforming is P=0,3, what’s the probability of having 10 non-conforming units if I will produce 20 units? If I look at the example graph in the image1, I have less than 0,05, which is less than 5% of probability.

Image1 - Binomial Distribution
Image1 – Binomial Distribution

In image1.1 you can look at the probability formula of the binomial distribution.

image1.1 - Binomial distribution formula
image1.1 – Binomial distribution formula

where:

  • n is the sample size;
  • x is the number of success;
  • p is the probability of success ;
  • q is the probability of non-success that is 1-p;


Poisson Distribution

The Poisson distribution is used when you don’t want to count the non-conforming unit, but you want to calculate the total number of non-conformity (because maybe one unit has more than one problem). In this case, you need to know the mean of the error on the other unit (Lamba, which is also equal to the variance) and the number of mistakes you want to occur on the next unit.


Example: if the mean of non-conforming on the other unit is 5, what probability has 0 non-conformities on the following units? If we look at the image1, the probability is near to 0.01, that is 1%
image2 - Poisson Distribution
image2 – Poisson Distribution

You can look at the probability formula of the Poisson distribution formula in the image2.1

 image2.1 - Poisson formula
image2.1 – Poisson formula

where:

  • Lamba is the mean;
  • x is the number of successes.


Hypergeometric distribution

The Hypergeometric distribution measures the probability of having a certain number of successes in extraction without replacement. In this case, you have two different outcomes, and each outcome of the experiment depends on the other. Although you want to measure the probability of success like the binomial distribution, the difference is that in the hypergeometric, the probability of success is not the same for each extraction.


Example: if you have 70units in your storage. Your storage has m=20 defective units and N-m=50 non-defective. If you randomly draw units from the storage without replacement, what's the probability of getting x=20 faulty units extracting n=30 elements? If we look at image3, it's near 0.10, which is 10%.
image3 – Hypergeomtric distribution

You can look at the probability formula of the Hypergeometric distribution in the image3.1

image3.1 - Hypergeometric formula
image3.1 – Hypergeometric formula

where:

  • N is the entire population size;
  • n is the number of unit extracted from the population;
  • m is the number of defective in the entire population;
  • x is the number of defective in the sample.


Remember that the combination formula is the one in image3.2

 image3.2 - Combination formula
image3.2 – Combination formula


Geometric distribution

The Geometric distribution is used when you have two outcomes for a trial, like conforming and non-conforming. It represents the number of failures before getting success. The outcome of each trial must be independent. A geometric distribution is like a succession of Binomial experiments.


Example: if your selecting a candidate for a work, and the probability of choose the correct one is p=0,5, what's is the probability after 2 job interview? if you loog at the image4 the probability is 0,25
image4 - Geomtric distribution
image4 – Geomtric distribution

In image 4.1, you can look at the formula of the probability of the geometric distribution where P is the probability of success and x is the number of tries:

image4.1 - Geometric distribution formula
image4.1 – Geometric distribution formula
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