Usually the day starts with a fresh brew that used to sit there half empty until the next day. But recently that is all

What’s particular about Monte Carlo simulations, is that the inputs are randomly generated. Continuous inputs are generated based on a probability density function, and discrete inputs based on a probability mass function.

When it’s your turn to play Risk and you choose to attack an enemy, you roll the dice. In 6-sided fair dice, you know you can only get a value between 1 and 6 for each die. You can’t get 2.3 on a die roll.

Try tossing a coin 5 times in a row. Maybe you’ll get all heads, or all tails. How sure are you that the probability of getting heads/tails is 1? Based on your 5 tosses it is, but we know that if they are truly independent tosses the probability would be close to 0.5. So there’s a lot of variance in the result from those 5 tosses.

If your random variable represents the toss of a fair coin, that function returns heads or tails with a probability of 1/2. But you have control over the random variable, and it can be anything that makes sense in the context of your problem.

With discrete random variables, you can ask questions about absolute probabilities, like what’s the probability that I get a 4 in the next roll? Or, if your opponent rolls the die, and gets a 3 and it’s your turn to attack, you might wonder what’s the probability of getting a value higher than 3?

But simpler methods include the one used by Buffon, where you have Bernoulli trials and record the probability associated with success and failure, success being heads and failure tails, or vice-versa.

The results obtained from performing a large number of trials should be close to the expected value. And it will become closer to the true expected value, the more trials you perform.

Running these simulations is a way to get all the pieces in place to use inferential statistics. The idea is to get lots outputs from the simulations, very much like you take random samples to study a population, and then use inferential statistics to draw conclusions about the actual phenomenon.

The model represents a probabilistic system, something that takes inputs and returns an output with a given probability. Everything depends on the model. If you don’t have a good model, you’ll have consistently poor results, Garbage in, Garbage Out.

For inputs that can be described using discrete random variables, you can only get results from a preset list of outcomes. So you use a Probability Mass Function (PMF) to get the probability of a particular outcome. It’s the equivalent of a PDF, but for discrete random variables.

Probability Density Functions (PDFs) describe the probability of each output of a continuous random variable. They give you a relative likelihood that the random variable will be equal to any given value. The interesting part is that it’s the relative probability, not an absolute one.

To run Monte Carlo Simulations you need a model, a way to represent the conditions of your experiment and how the different elements in the experiment interact (or not) to achieve a result or output.

You’ve noticed that since starting to work from home, you’ve had a harder time falling asleep at night, and even some episodes of insomnia. Although there may be several other factors, you quickly realize that you’ve just been having too much coffee altogether.

This aggregated output, can be the mean value for what you’ve modeled, like when you’re modeling temperatures, or the probability of a success/failure scenario, such as the likelihood of a team winning the Super Bowl.

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attempt to make the M1 machine my primary daily driver and development machine, be that web, Flutter or native. If it pans out as I hope it will, that leaves my 2018 MacBook Pro 15' available for trading in for potentially an Apple Silicon iMac or iMac Pro in 2021.