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
As an environmental scientist, you are studying the water quality of the Lanier watershed. Using historical records, you calculated the following probabilities:
- Probability of the water failing the nitrogen test: $P(N)=0.3$
- Probability of the water failing the phosphorus test: $P(P)=0.2$
- Probability of the water failing both tests: $P(N\cap P)=0.1$
What is the probability of the water failing the phosphorus test when you learned that it failed the nitrogen test $P(P|N)$?
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