# Probability

- 1 What is probability?
- 2 Epistemic vs. aleatory uncertainty
- 3 Deductive vs. inductive reasoning
- 4 Probability vs. statistics
- 5 What is the probability of a coin landing on heads?
- 6 Dice questions
- 7 Bayes’ theorem
- 8 Failing in math and/or science
- 9 Statistics in environmental science
- 10 Probability distribution
- 11 Reading materials
- 12 Homework: Conditional probability

## 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.

## 4 Probability vs. statistics

## 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.

## 11 Reading materials

- Probability
- Probability versus Statistics
- Probability vs Statistics
- What’s the difference between probability and statistics?
- The Difference Between Deductive and Inductive Reasoning
- Deductive Reasoning vs. Inductive Reasoning
- Can you say that statistics and probability is like induction and deduction?
- Bayes’ theorem
- Statistical concepts in environmental science
- Descriptive statistics
- Statistical inference
- Normal distribution
- Central limit theorem
- Standard Deviation and Variance

## 12 Homework: Conditional probability

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 $P(S|M)$?

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