Fundamentals of Probability in Statistics

Probability and Random Phenomena

Random Phenomenon

A random phenomenon is an event whose individual outcomes appear haphazard, but there is a regular distribution of outcomes when repeated many times. Examples include rolling a die or tossing a coin.

• Probability: The proportion of times an outcome can be expected to occur when repeated a large number of times (i.e., relative frequency).

Examples of Probabilities

• Rolling a die and getting a 5: $P(5) = \frac{1}{6} = 0.166... = 16.7\%$

• Tossing a coin and getting tails: $P(\text{tails}) = \frac{1}{2} = 50\%$

• Having a girl child: $P(\text{girl}) = \frac{1}{2} = 50\%$

Note: Probability is often studied in the context of gambling, such as poker, craps, and roulette.

Sample Space

The sample space is the set of all possible outcomes of a random phenomenon.

• Toss a single coin: $S = \{\text{heads}, \text{tails}\}$

• Toss two coins: $S = \{\text{HH}, \text{HT}, \text{TH}, \text{TT}\}$

• Roll a pair of dice and observe the sum: $S = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$

Event

An event is a subset of the sample space for a random phenomenon.

• Example: Drawing a heart from a deck of cards; rolling a 7 or 11 on a pair of dice.

Probability Model

A probability model assigns probabilities to events of a random phenomenon.

Basic Properties of Probability

• 1. The probability of an event will always be between zero and one: $0 \leq P(A) \leq 1$

• 2. The probability of all possible outcomes is one: $P(S) = 1$

• 3. If $P(A) = 0$, A cannot occur.

Assigning Probabilities

Classical Probability

If a sample space consists of a list of equally likely outcomes:

• $P(A) = \frac{\text{number of outcomes in A}}{\text{total number of outcomes in the sample space}}$

Empirical Probability

After a probability experiment is performed a large number of times, empirical probability can be found by:

• $P(A) = \frac{\text{number of outcomes where A did occur}}{\text{number of trials}}$

Example Table: Sums of Two Dice

The table below shows the possible sums when rolling two dice:

Dice 1	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Example: Probability of rolling a 7 or 11: $P(7 \text{ or } 11) = \frac{8}{36} = 2/9$

Venn Diagrams and Probability

Venn Diagram

A Venn diagram represents events and their probabilities visually.

• Example: Probability of owning a cat and/or dog among students. If 10% have both, 35% have only cats, and 30% have only dogs, the Venn diagram illustrates these relationships.

Properties of Probability

• 4. The probability that event A will not occur is one minus the probability that A does occur: $P(A') = 1 - P(A)$

• 5. If A and B have no outcomes in common, $P(A \text{ or } B) = P(A) + P(B)$

• 6. If A and B are any two events, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Example Calculation

• Probability of owning a cat or dog: $P(\text{cat or dog}) = P(\text{cat}) + P(\text{dog}) - P(\text{cat and dog}) = 0.45 + 0.4 - 0.1 = 0.75$

Types of Events

Independent Events

Events A and B are independent if knowing the result of one does not affect the result of the other.

• Example: Tossing heads on the first and second toss of a coin.

Dependent Events

Events A and B are dependent if the result of one does affect the result of the other.

• Example: Drawing an ace from a deck of cards, then drawing another card.

Conditional Probability

Conditional probability $P(A|B)$ is the probability that A will occur, given that B has occurred.

• $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

Example Table: Conditional Probabilities for Dice

Dice 1	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

• Example: $P(A|B) = \frac{4}{6} = 2/3$ if B restricts the sample space to outcomes 2, 3, 4, 6.

More Probability Properties

• 7. If events A and B are independent, $P(A \text{ and } B) = P(A) \cdot P(B)$

• 8. Events A and B are independent if $P(A|B) = P(A)$

• 9. If A and B are any two events, $P(A \text{ and } B) = P(A) \cdot P(B|A)$

• 10. $P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

Tree Diagram Example

A tree diagram can be used to represent the probabilities of sequential events, such as drawing balls from a box.

• Example: Probability of selecting 2 white balls from the box: $P(\text{white and white}) = P(\text{white 1st ball}) \times P(\text{white 2nd ball | white 1st ball}) = \frac{4}{9} \times \frac{3}{8} = \frac{12}{72} = \frac{1}{6}$

Summary Table: Key Probability Properties

Property	Formula
Complement Rule	$P(A') = 1 - P(A)$
Addition Rule (disjoint)	$P(A \text{ or } B) = P(A) + P(B)$
General Addition Rule	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
Multiplication Rule (independent)	$P(A \text{ and } B) = P(A) \cdot P(B)$
Conditional Probability	$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$

Additional info: These notes cover foundational probability concepts, including sample spaces, events, classical and empirical probability, Venn diagrams, independent and dependent events, conditional probability, and tree diagrams. The examples and tables are designed to reinforce understanding and provide practical applications for college-level statistics students.