Elements of Probability: Structured Study Notes

Chapter 3: Elements of Probability

3.2 Sample Space and Events

In probability theory, understanding the concepts of sample space and events is fundamental. These concepts form the basis for analyzing random experiments and calculating probabilities.

• Sample Space (S): The set of all possible outcomes of an experiment. For example, when determining the sex of a child, the sample space is S = {girl, boy}.

• Event: Any subset of the sample space. An event may consist of one or more outcomes. For instance, the event that a child is a girl is E = {girl}.

• Example: If an experiment consists of drawing a number from 1 to 6, the sample space is S = {1, 2, 3, 4, 5, 6}. The event that the number is 3 is E = {3}.

Additional info: Events can be simple (single outcome) or compound (multiple outcomes).

3.3 Venn Diagrams and the Algebra of Events

Venn diagrams are graphical tools used to visualize relationships between events in a sample space. They help illustrate unions, intersections, and complements of events.

• Union (E ∪ F): The event that either E or F or both occur.

• Intersection (E ∩ F): The event that both E and F occur.

• Complement (Ec): The event that E does not occur.

• Distributive Law: $E igcup (F igcap G) = (E igcup F) igcap (E igcup G)$

Operation	Symbol	Venn Diagram
Union	$E igcup F$	Shaded region covers both E and F
Intersection	$E igcap F$	Shaded region covers overlap of E and F
Complement	$E^c$	Shaded region outside E

3.4 Axioms of Probability

The axioms of probability provide the foundation for all probability calculations. They are based on the empirical observation of repeated experiments.

• Axiom 1: $0 ext{ ≤ } P(E) ext{ ≤ } 1$

• Axiom 2: $P(S) = 1$

• Axiom 3: For any sequence of mutually exclusive events $E_1, E_2, ...$, $Pigg(igcup_{i=1}^{ ext{∞}} E_iigg) = ext{∑}_{i=1}^{ ext{∞}} P(E_i)$

Proposition: $P(E^c) = 1 - P(E)$

3.5 Sample Spaces Having Equally Likely Outcomes

When all outcomes in a sample space are equally likely, probability calculations become straightforward.

• Probability of an event E: $P(E) = rac{n(E)}{n(S)}$ where $n(E)$ is the number of outcomes in E and $n(S)$ is the total number of outcomes in S.

• Basic Principle of Counting: If two experiments are performed, the total number of possible outcomes is the product of the number of outcomes for each experiment.

• Example: Drawing 3 bulbs from a box containing 6 white and 3 black bulbs. The probability that one drawn bulb is black is $P = rac{3}{9}$.

Permutations and Combinations

• Permutation: The arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is $P(n, r) = rac{n!}{(n-r)!}$

• Combination: The selection of objects without regard to order. The number of combinations of n objects taken r at a time is $C(n, r) = rac{n!}{r!(n-r)!}$

3.6 Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred.

• Definition: The probability of event E given event F is $P(E|F) = rac{P(E igcap F)}{P(F)}$

• Example: If a die lands on an even number, the probability that it is a 4 is $P(4|even) = rac{1}{3}$

3.7 Bayes' Formula

Bayes' formula allows for the updating of probabilities based on new information. It is especially useful in cases where events are not equally likely.

• Bayes' Theorem: $P(E_i|F) = rac{P(E_i)P(F|E_i)}{ ext{∑}_{j=1}^{n} P(E_j)P(F|E_j)}$

• Example: In medical testing, Bayes' theorem can be used to update the probability that a person has a disease given a positive test result.

3.8 Independent Events

Events are independent if the occurrence of one does not affect the probability of the other.

• Definition: Events E and F are independent if $P(E igcap F) = P(E)P(F)$

• Example: Tossing two coins. The outcome of one does not affect the other.

Properties of Independence

• If E and F are independent, then E and Fc are also independent.

• For three events E, F, G, independence requires $P(E igcap F igcap G) = P(E)P(F)P(G)$

Problems and Applications

The chapter concludes with a variety of problems that apply the concepts of sample spaces, events, probability axioms, conditional probability, Bayes' theorem, and independence. These problems reinforce understanding and provide practice in real-world scenarios.

• Example: Calculating the probability that a randomly selected couple earns less than $50,000 using tabular data.

• Example: Determining the probability of drawing certain colored balls from a box.

Additional info: Problems cover combinatorial arguments, Venn diagram analysis, and applications in genetics, engineering systems, and medical testing.