CH2 - Probability

Essence

Experiment: a process the results in an outcome that cannot be predicted in advance with certainty.

Toss a Coin

S = {H,T}

Sample space: set of all possible outcomes

Rolling a die

S = {1,2,3,4,5,6}

Event: subset of the sample space

Back to the examples:

Let even A be the roll of 1, A={1}

Let event E be an even role, E={2,4,6}

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Up to this point, you have been a “frequentist”

$$p(A)=\underset{n\to \infty }{\lim }\frac{\text{favorable outcomes}}{\text{total number of trials}}$$

Plugging in the numbers:

$$p(A)=\frac{1}{6}$$

Basic Axioms:

Axioms are the basic building blocks of probability theory.

Here are the three basic axioms:

P(a) is greater than or equal to 0, but less than or equal to 1 for all events a

$$0\le P(a)\le 1$$

P(S) is equal to 1

$$P(S)=1$$

P(A or B) = P(A) + P(B) - P(A and B)

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Ven Diagrams

Setting up the environment:

Python

import micropip await micropip.install("matplotlib") await micropip.install("matplotlib_venn") import matplotlib.pyplot as plt from matplotlib_venn import venn2 # Data labels = ['A', 'B', 'A and B'] plt.figure(figsize=(6, 4))

Output

1. Not in A

Python

venn2(subsets=(1, 0, 0), set_labels=labels) plt.title('Not in A') plt.show()

Output

2. Not in B

Python

venn2(subsets=(0, 1, 0), set_labels=labels) plt.title('Not in B') plt.show()

Output

3. A and B

Python

venn2(subsets=(0, 0, 1), set_labels=labels) plt.title('A and B') plt.show()

Output

4. A or B

Python

venn2(subsets=(1, 1, 1), set_labels=labels) plt.title('A or B') plt.show()

Output

5. DeMorgans Law

DeMorgan’s Law states that the complement of the union of two sets is equal to the intersection of the complements of the two sets.

Python

venn2(subsets=(1, 1, 0), set_labels=labels) plt.title("DeMorgan's Law") plt.show()

Output

Lets examine the ideas DeMorgans Law presents.

Quantification Vs Complement

Quantification	Complement
All 5	Not all 0, 1, 2, 3, 4
1,2,3,4,5	Not some

Are events mutually exclusive?

Depends on the problem.

Lets look at a deck of cards:

Deck of Cards

S = {52 cards} A = {red cards} B = {face cards} A and B = {red face cards}

Are A and B mutually exclusive?

No, because there are red face cards.

The intersection of A and B is not empty.