Probability

Dr. Huidae Cho
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.

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

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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