# Probability

Institute for Environmental and Spatial Analysis...University of North Georgia

## 1   What is probability?

We have to embrace uncertainty when studying science because we only have limited knowledge.

The lack of certainty or confidence is called uncertainty.

## 2   Epistemic vs. aleatory uncertainty

Epistemic uncertainty arises because of the lack of our knowledge.

Aleatory uncertainty arises because of randomness.

## 3   Deductive vs. inductive reasoning

Deductive reasoning starts with ideas or premises and observes data to make a conclusion.

Inductive reasoning starts with observations and analyzes data to formulate a theory.

## 5   What is the probability of a coin landing on heads?

Do you know this probability in advance without any experiments?

Do you have to throw a coin a lot of times to observe what happens?

## 6   Dice questions

• What is the probability of a die rolling a 1?
• What about a 1 and then a 6 in a sequence?
• A 1 and a 6 from two dice simultaneously?

## 7   Bayes’ theorem

$P(A|B) = \frac{P(A\cap B)}{P(B)} = \frac{P(B|A)P(A)}{P(B)}$

## 8   Failing in math and/or science

Probability of failing in math: $P(M)=0.3$

Probability of failing in science: $P(S)=0.2$

Are these two events related or independent?

Probability of failing in both math and science: $P(M\cap S)=0.1$

What is the probability of failing in either math or science $P(M\cup S)$?

What is the probability of failing in science when you learned that you failed in math $P(S|M)$?

## 9   Statistics in environmental science

Descriptive statistics is used to describe data.

Inferential statistics is used to make predictions.

## 10   Probability distribution

Statisticians and probabilists love normal distributions thanks to the central limit theorem.

$f(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$ where

• $x$ is a random variable,
• $\mu$ is the mean or expected value of $x$, and
• $\sigma$ is the standard deviation.

Probability of failing in math: $P(M)=0.3$
Probability of failing in science: $P(S)=0.2$
Probability of failing in both math and science: $P(M\cap S)=0.1$
What is the probability of failing in science when you learned that you failed in math $P(S|M)$?