Probability Study Notes

Chapter 5: Probability

The Concept of Probability

• Definition: The idea of probability is based on the concept that random behavior can be predictable in the long run.
• In Section 5.1, simulation was used to imitate random processes (e.g., tossing coins, rolling dice).
• Key Question: Do we need to repeat a random process multiple times to determine the probability of a specific outcome?
  • Answer: No, we do not always need to repeat the random process multiple times.

Probability Models

• In previous chapters, different types of modeling techniques were introduced:
  • Chapter 2: Normal density curves modeled distributions of quantitative data.
  • Chapter 3: Linear relationships between two quantitative variables were modeled using least-squares regression lines.
• Development of Probability Models: Now we focus on developing models for random behavior.
• Example: Rolling two fair six-sided dice (one red, one blue).
  • Sample Space: There are 36 possible outcomes when rolling these two dice. Each outcome is equally likely, with a probability of $\frac{1}{36}$.

Definitions

• Probability Model: A description of a random process comprised of:
  • A list of all possible outcomes.
  • The probability for each outcome.
• Sample Space: The collection of all possible outcomes of a random process.
• Example of Sample Space: For a single coin toss, the sample space is {Heads, Tails}. In a survey of 1523 U.S. adults, the sample space consists of all possible sets of responses from these adults.

Events and Probabilities

• Definition of an Event: A collection of outcomes from a random process, usually denoted by capital letters (e.g., A, B, C).
• Example of Event: For the dice example, let event A be defined as getting a sum of 5.
  • Probability notations: The probability of event A is denoted as P(A) or P(sum is 5).
• Number of outcomes in event A corresponding to a sum of 5 is 4 outcomes.
• Probability Calculation: The probability of event A is given by:
  $P(A) = \frac{\text{Number of outcomes in event A}}{\text{Total number of outcomes in sample space}} = \frac{4}{36} = 0.111.$

Finding Probabilities with Equally Likely Outcomes

• General Formula: If all outcomes in the sample space are equally likely, use:
  $P(A) = \frac{\text{Number of outcomes in event A}}{\text{Total number of outcomes in sample space}}.$
• Example Problem: Spin the Spinner
  • (a) Define a probability model for a spinner with three equal sections: red, blue, and yellow.
  • (b) Define event A as spinning blue at least once and find P(A).
  • Solution: Sample Space = {RR, RB, RY, BR, BB, BY, YR, YB, YY}.
  • Each outcome has a probability of $\frac{1}{9}.$
  • Event A (spinning at least one blue) = {RB, BR, BB, BY, YB}.
  • Therefore, P(A) = $\frac{5}{9} = 0.556$.

Basic Probability Rules

1. Rule 1: For equally likely outcomes, the probability of event A can be calculated as:
   $P(A) = \frac{\text{Number of outcomes in event A}}{\text{Total number of outcomes in sample space}}.$
2. Rule 2: The probability of any event is between 0 and 1, derived from the definition of probability as a proportion.
3. Rule 3: The sum of the probabilities of all possible outcomes in a sample space equals 1.
   • Example: Probability of results for a sum of rolling two dice - Probability of not getting a sum of 5:
     $P(sum is not 5) = 1 - P(sum is 5) = 1 - \frac{4}{36} = \frac{32}{36} = 0.889.$
4. Complement Rule: Defines that the probability an event A does not occur is given by:
   $P(A^C) = 1 - P(A)$, where A^C represents the complement of A.

Mutually Exclusive Events and Addition Rule

• Definition of Mutual Exclusivity: Two events A and B are mutually exclusive if they cannot occur at the same time: $P(A and B) = 0.$
• Addition Rule for Mutually Exclusive Events:
  • If A and B are mutually exclusive:
    $P(A or B) = P(A) + P(B).$
  • Example: Finding the probability of getting a sum of 5 or 6 when rolling dice:
    $P(sum is 5 or 6) = P(sum is 5) + P(sum is 6) = \frac{4}{36} + \frac{5}{36} = \frac{9}{36} = 0.25$ (events share no common outcomes).

Summary of Basic Probability Rules (Symbolic Form)

• General constraints:
  • For any event A, $0 \leq P(A) \leq 1$.
  • If S is the sample space, then $P(S) = 1$.
  • If outcomes are equally likely:
    $P(A) = \frac{\text{Number of outcomes in event A}}{\text{Total number of outcomes in sample space}}.$
  • Complement rule: $P(A^C) = 1 - P(A)$.
  • Addition rule for mutually exclusive events:
    $P(A or B) = P(A) + P(B).$

Example of Non-Equally Likely Outcomes

• Problem: M&M's probability model for different colors:
  • Probability assignment:
  • Blue: 0.207
  • Orange: 0.205
  • Green: 0.198
  • Yellow: 0.135
  • Red: 0.124
  • Brown: 0.131
• Check validity: Each probability lies between 0 and 1, their sum: 1.
• Find non-blue M&M probability:
  • $P(not blue) = 1 - P(blue) = 1 - 0.207 = 0.793$.
• Calculate orange or brown M&M probability:
  • $P(orange or brown) = P(orange) + P(brown) = 0.205 + 0.124 = 0.329$ (addition rule applies as they are mutually exclusive).

Introduction to Two-Way Tables and General Addition Rule

• For two events that are not mutually exclusive, a Two-Way Table can be effective for probability calculation.
• Example survey for Facebook and Instagram usage in residents:
  • Outcomes: 68% use Facebook, 28% use Instagram, 25% use both.
  • Probability of using either platform is calculated using the General Addition Rule:
    $P(A or B) = P(A) + P(B) - P(A and B).$
  • Solve for combined probability:
    $P(F or I) = P(F) + P(I) - P(F and I) = 0.68 + 0.28 - 0.25 = 0.71.$

Summary of General Addition Rule

• The general addition rule accounts for overlapping probabilities among non-mutually exclusive events and is a critical aspect of probability modeling.

Venn Diagrams and Probability

• Definition: A Venn diagram visually represents events and their relationships.
  • Each circle represents an event and is contained within a rectangle indicating the sample space.
• Notation for relationships in probability:
  • Intersection (A and B): Denotes outcomes common to both events, indicated by $(AB)$.
  • Union (A or B): Denotes all outcomes in either event, indicated by $(AUB)$.
• General Addition Rule can be symbolically rewritten as:
  $P(AUB) = P(A) + P(B) - P(AB).$

Conclusion

• The probability model provides a fundamental framework for understanding random processes by integrating outcomes and their respective probabilities, summarizing data through rules and principles essential for analysis. This framework extends to cover intersection, union, and non-mutually exclusive scenarios effectively using tables and graphical illustrations.

Section 5.2 Exercises

Exercise Samples:

1. Dice Model: Roll two four-sided dice, define event A as getting a sum of 5.
2. Coin Tosses: Model for tossing a coin three times, define event B as more heads than tails.
3. Grandkid Selection: Randomly select 2 out of 6 grandchildren, find probability that at least one girl gets to go.
4. M&M's Probability: Model of winning prizes states valid probabilities and calculates expected outcomes for candy draws.
5. Cholesterol Levels: Define events for different cholesterol categories, apply probability definitions to assess outcomes.