W8- OB notes PART 1

6-1 Assigning Probability to Events

Random Experiments

• Definition: A random experiment is an action that can produce one of several outcomes.

• Requirements: All possible outcomes must be exhaustive (cover every possibility) and mutually exclusive (no overlap between outcomes).

Examples:

• Flipping a Coin: The outcomes are Heads or Tails. Each flip has an equal chance (1/2) resulting in either outcome.

• Test Scores: When recording marks on a test, the possible outcomes range from 0 to 100.

• Grading Scheme: Possible grades could be A, B, C, D, or F, illustrating different performance levels in an evaluation.

• Rolling a Die: The outcomes of rolling a six-sided die are 1, 2, 3, 4, 5, or 6. Each number has an equal probability of 1/6.

• Drawing a Card: The outcome of drawing a single card from a standard deck can be any of the 52 cards, including 13 different ranks of each suit.

Sample Space and Probability Requirements

• Sample Space: Denoted as S, it is the complete set of all possible outcomes. For instance, flipping a coin has a sample space of S = {Heads, Tails}.

• Probability Requirements:

  • Each probability assigned to an outcome must range from 0 to 1 (0 ≤ P(i) ≤ 1).

  • The total probability across all outcomes must equal 1 (Σ P(i) = 1).Example: In the case of a fair coin, the probability for Heads is P(Heads) = 0.5 and for Tails is P(Tails) = 0.5. Summed, this gives ⟹ 0.5 + 0.5 = 1.

Approaches to Assigning Probabilities

1. Classical Approach: This method assumes outcomes are equally likely.

   • Example: The probability of rolling a 3 on a fair die is P(3) = 1/6, as each side is equally likely.

2. Relative Frequency Approach: This is based on empirical data from experiments or observations.

   • Example: If a die is rolled 100 times and a 3 appears 15 times, the empirical probability would be P(3) = 15/100 = 0.15.

3. Subjective Approach: When there is no statistical data, personal judgment is used.

   • Example: A manager may estimate the likelihood of a project succeeding as 70% based on experience and intuition.

Defining Events

• Simple Event: Represents a single outcome from the sample space.

  • Example: Rolling a 4 on a die is a simple event.

• Event: A collection of one or more simple events.

  • Example: To score an A in a statistics class may include the range of scores from 80 to 100, expressed as E = {80, 81, 82, …, 100}.Additional Examples:

• Rolling an Even Number: The event of rolling an even number on a die is expressed as E = {2, 4, 6}.

• Drawing a Heart: The event of drawing a heart from a deck consists of 13 outcomes: E = {2 of Hearts, 3 of Hearts, ..., King of Hearts}.

Probability of an Event

To determine the probability of an event, we sum the probabilities of all constituent simple events.Example: If P(A) = 0.20 (event A), P(B) = 0.30 (event B), P(C) = 0.25 (event C), and P(D) = 0.15 (event D), then the total probability of passing (P(Pass)) aggregates as follows: P(Pass) = P(A) + P(B) + P(C) = 0.2 + 0.3 + 0.25 = 0.75. In this aggregation, we see that events can overlap and each contributes to the overall probability. Additional Example: If the probabilities for events A, B, and C were given as 0.5 (win), 0.4 (lose), and 0.1 (draw), it confirms that the outcome probabilities total 0.5 + 0.4 + 0.1 = 1.0, allowing for clear decision-making based on probabilities.

Interpreting Probability

Probability interpretations can align closely with empirical findings.Example: If a stock research analyst estimates a 65% chance for a stock's value to increase, it implies that among many stocks with similar characteristics, approximately 65% would appreciate in value when analyzed over extended periods.

6-2 Joint, Marginal, and Conditional Probability

Intersection

The intersection of two events A and B, represented as A ∩ B, refers to the event where both events occur. For example, if event A is rolling an even number and event B is rolling a number greater than 3, then the intersection A ∩ B yields the numbers {4, 6} as both meet the criteria.

Marginal Probabilities

These probabilities are computed by summing joint probabilities across a relevant dimension, like rows or columns in probability tables.Example: In a study involving mutual fund managers, the marginal probability P(A1) can be determined as:P(A1) = P(A1 ∩ B1) + P(A1 ∩ B2).Utilizing two-way tables can help provide clarity and assist in visualizing how these probabilities relate.

Conditional Probability

This refers to the probability of event A, given that event B has occurred, denoted as P(A|B) = P(A ∩ B) / P(B).Example: The probability that a fund managed by a top-20 MBA program performs well could be analyzed given its managerial pedigree, offering insight into performance metrics of those specific funds.

Independence

Two events A and B are said to be independent if the occurrence of one does not influence the other; thus, P(A|B) = P(A).Example: The probability of rolling a 3 on a die remains independent of flipping a coin.

Complement Rule and Union of Two Events

The complement of an event A, denoted A^C, occurs when A does not take place.Example: If P(A) = 0.4, then P(A^C) = 1 - P(A) = 0.6, signifying all scenarios not encompassed by event A occur.Union: The union of events A and B signifies that at least one of them occurs, calculated as P(A ∪ B) = P(A) + P(B) – P(A ∩ B). Example: To determine the probability of drawing either a King or a Heart, remember that some cards serve as both a King and a Heart (1 out of 52), warranting overlap consideration.

Additional Cases

In complex modeling, events may relate intricately, exemplifying how union probabilities are articulated in case studies.

Probability Trees

Probability trees depict events visually through branches which represent conditional probabilities and joint outcomes.Use Case: Two-colored marbles drawn from a bag might illustrate various branching paths effectively, helping visualize how probabilities interact through successive draws.

6-3 Probability Rules and Trees

Multiplication Rule

This rule aids in calculating joint probabilities, especially when focusing on independent events for simplification.

Example with Students

Example scenarios can be crafted to choose students with and without replacement to demonstrate the impact of these calculations on availability.Example: Select 2 students from a group of 30; choosing one without replacement shifts the total potential choices in subsequent turns.

Addition Rule

Defines the probability of at least one of the events occurring as: P(A ∪ B) = P(A) + P(B) – P(A ∩ B).Example: A city newspaper subscription case may show how to apply the addition rule to ascertain total subscriptions effectively.

Application of Trees

Provide examples on organizing and calculating probabilities with trees, highlighting intricate scenarios like the success rate on a bar exam efficiently, fostering understanding of interdependencies between outcomes.

6-4 Bayes’s Law

Overview

Bayes’s Law recalibrates prior probabilities based on emerging evidence, offering updated perspectives post-event.

Example Applications

Applications of Bayes’s Law in settings like MBA programs can help interpret likelihoods effectively using probabilistic reasoning methodologies when tackling case studies.

Bayes’s Law Formula

An algebraic representation assists in computing complex conditional probabilities effectively, offering clarity to potentially convoluted relations.

6-5 Identifying the Correct Method

This section outlines necessary steps to ascertain if joint probabilities are provided or required, along with methods to compute relevant probabilities effectively.

Chapter Summary

Highlights key takeaways on assigning probabilities, various computation methods (like Bayes’s Law), the importance of conditioning assessments, evaluations of independence, and the applications of probability rules across different scenarios.