probability Archives - paologironi blog

The Monte Carlo Method Explained Simply with Real-World Applications

Monte Carlo simulation is a method used to quantify the risk associated with a decision-making process. This technique, based on random number generation, is particularly useful when dealing with many unknown variables and when historical data or past experiences are not available for making reliable predictions.

The core idea behind Monte Carlo simulation is to create a series of simulated scenarios, each characterized by a different set of variables. Each scenario is determined by randomly generating values for each variable. This process is repeated many times, thus creating a large number of different scenarios.

The Hypergeometric Distribution

We have seen that the binomial distribution is based on the hypothesis of an infinite population N, a condition that can be practically realized by sampling from a finite population with replacement.

If this does not occur, meaning if we are sampling from a population without replacement, we must use the hypergeometric distribution. (In reality, if N is large, the hypergeometric probability density function tends towards the binomial).

The hypergeometric distribution is used to calculate the probability of obtaining a certain number of successes in a series of binary trials (yes or no), which are dependent and have a variable probability of success.

The hypergeometric distribution allows us to answer questions like:

If I take a sample of size N, in which M elements meet certain requirements, what is the probability of drawing x elements that meet those requirements?

The Negative Binomial Distribution (or Pascal Distribution)

The negative binomial distribution describes the number of trials needed to achieve a certain number of successes in a series of independent trials. For example, it could be used to calculate the probability of getting three heads when flipping a coin 5 times, assuming the coin is balanced and therefore the probability of getting heads on each flip is 50%.

The negative binomial distribution is useful in many fields, including statistics, economics, biology, and physics. And also in “our” SEO.

First Steps into the World of Probability: Sample Space, Events, Permutations, and Combinations

Probability and combinatorics are two fundamental concepts in mathematics and statistics that help us understand and interpret many phenomena in everyday life. In this introductory post, we’ll “touch upon” the main concepts together, seeing how they can be applied in various contexts.