CH1 - Randomness

Randomness refers to the inherent unpredictability of a sequence of events. In statistics, randomness is a key concept that underlies many statistical methods and procedures. Understanding randomness is essential for interpreting data and making informed decisions based on statistical analyses.

In this chapter, we will examine some concepts of randomsness: outcomes, events, and sample spaces.

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Outcomes, Events, and Sample Spaces

Look at the experiements/trials:

$$SampleSpace:S=\{setofallpossibleoutcomes\}$$ $$RollingDie:\to \{1,2,3,4,5,6\}$$

An event is a collection of some outcomes,

a subject of the sample space.

Example

Let $D$ be the outcome of rolling the die and getting a 1:

$$A=\{1\}$$

Let $E$ be the event of rolling an even number.

$$E=\{2,4,6\}$$

Let $D$ be the event of rolling a number divisible by 3.

$$D=\{3,6\}$$

Venn Diagram

Where every outcome $A$ is also an outcome of $B$ inside sample space $S$

Python

import matplotlib.pyplot as plt fig, ax = plt.subplots() sample_space = plt.Rectangle((-1.5, -1.5), 6, 6, fill=None, edgecolor='black', lw=2, label='S (Sample Space)') ax.add_patch(sample_space) circle_B = plt.Circle((2, 2), 2.5, color='blue', alpha=0.3, label='B') ax.add_patch(circle_B) circle_A = plt.Circle((2, 2), 1.5, color='red', alpha=0.5, label='A') ax.add_patch(circle_A) plt.text(2, 2.9, 'B', color='blue', fontsize=12, ha='center') plt.text(2, 1.4, 'A', color='red', fontsize=12, ha='center') plt.text(-1.4, 5.4, 'S (Sample Space)', fontsize=10) ax.set_xlim(-2, 4) ax.set_ylim(-2, 4) ax.set_aspect('equal') plt.title('Venn Diagram: A ⊆ B ⊆ S') plt.legend(loc='upper right') plt.show()

Output

For brevity, won’t include Venn diagram for each relation, but I will list a few of them:

Relations of Events

And their included notation

De Morgan’s Laws

Notation:

$$A\cup B=A^{\prime }\cap B^{\prime }$$

Abstract:

De Morgan’s Laws are a pair of fundamental rules in set theory that describe how the union and intersection of sets are related. The first law states that the complement of the union of two sets is equal to the intersection of the complements of the sets. The second law states that the complement of the intersection of two sets is equal to the union of the complements of the sets.

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Applying What We Know So Far

> Roll 1 die twice.

• Let $X$ be the $1^{st}$ roll
• Let $Y$ be the $2^{nd}$ roll

Make a table

11	12	13	14	15	16
21	22	23	24	25	26
31	32	33	34	35	36
41	42	43	44	45	46
51	52	53	54	55	56
61	62	63	64	65	66

This yields 36 possible outcomes.

Let $X+Y$ be the sum of the 2 rolls:

$$S=\{2,3,4,5,6,7,8,9,10,11,12\}$$

We let $F$ be the event that the sum of the 2 rolls equals 4.

$$F=\{31,22,13\}=\{XY|X+Y=4\}$$

Let $T$ be the event that $Y=3$

$$T=\{XY|Y=3\}=\{13,23,33,43,53,63\}$$

Let $G$ be the event that $X$ is greater than $2Y$.

$$G=\{XY|X>2Y\}=\{31,41,51,61,52,62\}$$

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Playing Angry Birds

Say you are playing the famous and fun game Angry birds. Let’s say that for this case:

• $W$ = win
• $L$ = lose

Every time you lose, you have to start over from the beggning.

Consider the event that you win in less than $1000$ times:

Here, we can see that the sample space is infinite, and the event that you win in less than $1000$ times is a finite event.