### How it Works

• The base rate (Bayesian prior) probability is 0.85 because 85% of the cars are green.
• The hit rate is 0.80 because the reliability of the witness is 80% correct identification.
• The false alarm rate is 0.20 because the witness is incorrect 20% of the time.
• Bayes formula: ``` (x*y) / (x*y) + (z*1 - z*x) x = base rate y = hit rate z = false alarm rate ```

If we plug these variables into the calculator we see the probability the hit and run cab was blue is only 41%. Thus, Even though the witness is correct 80% of the time the actually probability the cab was blue is 41%. The theorem results are most influenced and weighted by the base rate (prior probability). The result is more likely to be near the base rate. This is true even if an evidence event seems to be high proof of the hypothesis. A base rate of 1% will result in 3.88% probability of hypothesis with a hit rate at 80%. If a base rate probability is less than 1%, even with a 90% hit rate probability, the result probability will only be 8.3%. It will take many new evidence events to even increase the hypothesis probability to greater than 50%.

If the result hypothesis is entered as the new base rate, and the other variables remain the same, one can get an intuition of how the probability of the hypothesis becomes more certain given the mounting evidence. The base rate must be as accurate, and relevant to the real life situation, as possible. If one doesn't know the hit rate, or false alarm rate, estimates can be supplied for these variables.

### What is Bayesian Probability?

Bayesian probability is a mathematical theory that allows us to calculate the probability of an event.

Bayesian statistics has been around for nearly 200 years and has been gaining momentum in recent years. It is now used by many businesses and organizations as a way to make better - less biased - decisions.

A Bayesian decision rule is a mathematical calculation that produces an estimate of the best decision given all the available information about what might happen.

### How Bayesian Probability Calculations Actually Work

Bayesian probability calculations are used to calculate the probability of a given event. They are typically used in decision-making and data analysis.

Bayes' theorem is the mathematical formula that determines the probability of an event happening, given some other information about it.

### The Importance of Using a Bayesian Probability

The Bayes theorem is a useful tool for making decisions, and for optimizing your predictive models.

The Bayes theorem is a very simple and powerful rule that can be used to calculate the probability of an event happening or not happening. It can be applied in many different situations such as credit risk, marketing, medical diagnosis, and so on.

The Bayes theorem is also known as Bayesian probability theory. It was published by Thomas Bayes in 1763 as a result of his study on data analysis to improve the accuracy of predictions about future events based on previous knowledge.

### The History of Bayes' Theorem's Birth and Evolution

Bayes' theorem is a statistical formula that was first published in 1763 by English mathematician Thomas Bayes. It states that the probability of an event can be inferred from the data about it.

Bayes' theorem was so influential because it introduced probability to statistics and how it can impact decisions. It also made calculations easier to do, which led to the rise of probabilistic thinking in science and society at large.

### The Effectiveness of Bayesian Analysis in the Criminal Justice System

Bayesian analysis is a strategy in forensics that helps to interpret crime-solving evidence. It’s based on the use of probabilities and it’s considered among the most effective strategies in criminal justice.

Bayesian analysis for criminal justice is a method used to ensure that crime solutions are based on accurate evidence and realistic expectations. It’s also known as "the science of reasonable doubt."

The Bayesian analysis is a meta-thinking methodology often used in the context of problem solving and decision-making. The methodology goes beyond simply determining how probable it is that a certain outcome will take place, and makes sense of how other factors influence outcomes, such as the probability of an event taking place or not taking place, or other individual events occurring or not occurring.