Prob & Stats: 5.1 + Ch. 5
Unit 2: Probability
Plan for Today
Introduction to Unit 2: Starts with a deep dive into probability.
Overview Topics:
Consideration of probability distributions.
Focus on special types of distributions.
Word of Caution: Understand the material carefully.
Today's Focus: Chapter 5, specifically Section 5.1, along with additional discussions from the rest of Chapter 5.
What is Probability?
Definition:
"The likelihood of a random process resulting in a specific outcome."Interpretation:
Practically, probability can be viewed as the proportion of times a certain outcome occurs over an extended series of trials.
The Law of Large Numbers
Principle:
As the number of trials (n) increases, the observed proportion of a specific outcome will converge to its actual probability.
Explanation: While the concept might sound complex, it is straightforward.
Example:
An example often cited is tossing coins.
Discussion Tool:
Use of StatCrunch applets for visual representation and further exploration of the Law of Large Numbers.
Foundations of Probability
Probability Experiment:
Defined as any repeatable process that yields uncertain results.
Probability (p):
Represents the chance that the outcome of the experiment will occur.
Key Terms:
Sample Space (S): The set of all possible outcomes.
Events (E or ei): Specific outcomes or sets of outcomes.
Example in Probability:
Consider the action of tossing a fair die, detail the sample space S, specify events ei(s), and assign appropriate probabilities to these events.
Probability Rules
Value Range:
Probabilities must be between 0 and 1, both inclusive.
Special Cases:
P = 0 refers to impossible events, while P = 1 denotes certain events.
Summation Rule:
The sum of probabilities of all possible outcomes must equal 1, expressed as:
S = {e1, e2, \ldots, en} \Rightarrow \sum P(ei) = P(e1) + P(e2) + \ldots + P(e_n) = 1
Uncertainty Requirement:
Probability is defined in contexts where the outcome is uncertain.
Probability Models
Definition:
A probability model comprises a listing of all possible outcomes along with their corresponding probabilities.
Interpreting Probabilities
Interpretation:
A probability value closer to 1 indicates a higher likelihood of occurrence.
Example applied: Weather forecasting for rain probability.
Expectations from Trials:
Through repeated trials, we should expect the actual results to align with theoretical probabilities.
Practical Issues:
Possible disturbances such as having a biased or weighted coin.
Relevance of gambling concerns.
Relation to Law of Large Numbers:
All interpretations return to the concept of the Law of Large Numbers.
Unusual Events
Definition of Unusual:
An event is termed "unusual" if its probability of occurrence is significantly low.
Cutoff Example:
A common threshold is $p = 0.05$, or 5%, suggesting that events falling below this standard may be considered unusual.
Flexibility of Cutoffs:
The 5% threshold is not absolute; various situations may require adjusted cutoffs.
Empirical Probability
Calculation Method:
Derived from observed results to estimate probabilities, essentially finding the relative frequency of outcomes.
Nature of Estimates:
This estimation stems from real data (known as empirical evidence).
Duration of observations tends to show variation, which aligns with the Law of Large Numbers.
Example of Empirical Probability
Survey Example:
Collected responses from 100 students at the College of Charleston regarding their living arrangements:
| Living Situation | Frequency | Probability |
|-------------------|-----------|-------------|
| On-campus | 39 | 0.39 |
| Off-campus | 43 | 0.43 |
| With parents | 18 | 0.18 |
Classical Probability
Definition:
Unlike empirical probability which is data-dependent, classical probability uses theoretical approaches based on counting techniques.
Focus:
It examines what is expected to happen over numerous trials.
Equally Likely Outcomes Requirement:
For calculations, each outcome must have an equal chance of occurring.
Calculating Classical Probability
Formula:
If a total of n possible outcomes exists and the event E can occur in m distinct ways, the probability is given by:
P(E) = \frac{m}{n}
Intuition:
The formula is more intuitive in its application than it initially appears.
Examples:
Rolling a fair die with the event of landing an even number.
Selecting a nursing major randomly from the class.
More Complex Examples
Two Dice:
Calculation methods include using tables to simplify the sample space.
Tree Diagrams:
Visualization for calculating probabilities, especially for multi-step events (e.g., example with triplets).
Outcome Retrieval:
Using tree diagrams aids in understanding the probability of various events occurring.
Empirical vs. Classical Probability
Comparison:
Empirical probability relies on actual data while classical probability is grounded in theoretical principles.
Variation Potential:
The two classifications seldom yield identical results, and this is an acceptable reality.
Implications of Discrepancies:
Notable differences, particularly as sample size (n) increases, merits discussion.
Subjective Probability
Definition:
The type of probability derived from personal judgment rather than strict mathematical inference.
Expert Input:
Ideally, estimates should come from individuals with considerable experience in the relevant field for credibility.
Legitimacy:
Although it can be viewed as less reliable, subjective probability can be valid and appropriate depending on the context.
Complements
Definition of Complement:
The complement of an event is the group of outcomes in the sample space that do not encompass the event itself.
Denotation:
Represented as Ec, the complement rule is defined as:
P(E^c) = 1 - P(E)
Example Calculation:
Roll a fair die; let E be the event of rolling a 1 or 6, then determine:
Sample Space (S)
The event E
The complement Ec
The probability P(E)
The probability of complement P(Ec)
Independent Events
Independent Definition:
Two events are independent if the occurrence of one does not affect the probability of the other occurring.
Dependent Events:
Conversely, if one event influences the probability of another, the events are considered dependent.
Examples for Analysis:
Tossing two coins and examining results.
Analyzing the occurrence of having two children.
Selecting 5 cards from a deck, observing variations depending on whether selection is with or without replacement.
Counting Techniques
Objective:
Calculate the total possible outcomes using specific counting techniques.
Focus on Particular Methods:
Combinations: Choosing r items from n options without regard to order and without replacement.
Permutations: Choosing r items from n options where order bears significance and without replacement.
Combinations
Combinatorial Formula:
To find how many ways to select r items from n, we utilize combinations:
C(n,r) = \frac{n!}{r!(n-r)!}Factorial Concepts:
Example: $5! = 5 * 4 * 3 * 2 * 1 = 120$
By definition, $0! = 1$
Usage in StatCrunch:
Navigate: Data > Compute > Expression; use
comb(n,r).
Permutations
Permutational Formula:
For selecting r items from n where order matters and no replacement occurs:
P(n,r) = \frac{n!}{(n-r)!}Exploration in StatCrunch:
Navigate: Data > Compute > Expression; utilize
perm(n,r).
Examples of Counting Techniques
Dessert Scenario:
If there are 7 people at a table, determine how many ways to choose 2 people for free dessert.
Committee Arrangement:
If there are 8 people on a committee, assess the number of ways to assign the positions of President, VP, and Secretary.
Number Selection:
If 50 numbers exist in a hat, find the number of ways to select 5 winning numbers.
Competition Rankings:
For 25 competitors, ascertain the possible arrangements for first, second, and third place finishes.