Exhaustive Probability Study Notes: Independence, Conditional Probability, and Multiplication Rules

Independence vs. Dependence of Events

Definition of Independent Events:
- Two events, $E$ and $F$ , are defined as independent if the occurrence of one event does not affect or change the probability of the occurrence of the other event.
- Example 1 (Fair Dice): Rolling a standard fair die to determine the probability of getting a number greater than 1. The outcomes greater than 1 are 2, 3, 4, 5, and 6, yielding a probability of $\frac{5}{6}$ . If two dice are rolled sequentially, rolling a 4 on the first die does not alter the probability of rolling a number greater than 1 on the second die; it remains $\frac{5}{6}$ . Rolls of fair dice are always independent events.
- Example 2 (Fair Coin Flips): Flipping a fair coin repeatedly produces independent events because previous flip outcomes do not alter future flip probabilities.
Definition of Dependent Events:
- Two events are considered dependent if the occurrence of one event affects or changes the probability of the occurrence of the other event.
- Example (Standard Deck of Cards):
  - A standard deck of cards contains 52 cards total, with no jokers. It consists of four suits: clubs, spades, diamonds, and hearts. Each suit contains 13 rank cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King.
  - The probability of drawing a Queen at random from a full standard deck is $\frac{4}{52}$ .
  - If a King is drawn first and removed from the deck (without replacement), the probability of subsequently drawing a Queen changes to $\frac{4}{51}$ .
  - Because the occurrence of drawing a King changed the probability of drawing a Queen, these two draws represent dependent events.

Multiplication Rule for Independent Events

Mathematical Formula for Independent Events:
- If $E$ and $F$ are independent events, the probability of both $E$ and $F$ occurring simultaneously is: $P(E \text{ and } F) = P(E) \times P(F)$
Key Word Indicators in Probability Problems:
- The word "and" serves as the indicator that the multiplication rule must be used.
- The word "or" serves as the indicator that the addition rule must be used.
Application Example 1 (60-Year-Old Female Survival Rate):
- Data: According to the National Vital Statistics Report, the probability that a randomly selected female aged 60 years old will survive the year is $99.186\text{\%}$ ( $0.99186$ ).
- Problem: Find the probability that two randomly selected 60-year-old females will survive the year.
- Rewording: Determine the probability that the first female survives and the second female survives.
- Assumption of Independence: Because the data reflects national population statistics across millions of individuals, one individual passing away does not change the probability of another individual passing away. They are assumed to be independent.
- Calculation: $P(\text{Female 1 survives and Female 2 survives}) = 0.99186 \times 0.99186 = 0.99186^2 \text{ or } 98.378\text{\%}$
Application Example 2 (Exercise Equipment Warranty Claims):
- Data: An exercise equipment manufacturer knows that $10\text{\%}$ ( $0.10$ ) of their products are defective. They also know that only $30\text{\%}$ ( $0.30$ ) of customers will actually use the equipment in the first year after purchase. The equipment comes with a 1-year warranty.
- Problem: What proportion of customers will make a valid warranty claim?
- Condition for Valid Claim: A customer makes a valid claim if the product is defective and the customer uses the product within the 1-year warranty period to discover the defect.
- Assumption of Independence: A customer purchasing the equipment does not know if it is defective at the time of purchase; defectiveness does not influence the customer's decision to use the equipment.
- Calculation: $P(\text{Defective and Used}) = P(\text{Defective}) \times P(\text{Used}) = 0.10 \times 0.30 = 0.03$
- Implication: Although $10\text{\%}$ of products are defective, the manufacturer only needs to replace or honor claims on $3\text{\%}$ ( $0.03$ ) of total sales.
Extension to $n$ Independent Events:
- The multiplication rule applies to any number of independent events: $P(E_1 \text{ and } E_2 \text{ and } \text{\dots} \text{ and } E_n) = P(E_1) \times P(E_2) \times \text{\dots} \times P(E_n)$
- Coin Flips Example: The probability of getting heads on 1 flip is $0.5$ . The probability of getting 3 heads in 3 consecutive coin flips is: $P(3 \text{ Heads}) = 0.5 \times 0.5 \times 0.5 = 0.5^3 = 0.125$
  - For 15 heads in 15 flips, the calculation is shorthand expressed as $0.5^{15}$ .
- Four Females Survival Example: The probability that four randomly selected 60-year-old females survive the year is: P(\text{All 4 Survive}) = 0.99186^4 \text{\approx} 0.9678
- Trend Observation: As more individuals are included in the sample group, the joint probability that every single individual in the group survives continuously decreases ( $0.99186 \rightarrow 0.9678$ ).

Calculating Probabilities of Complements ("At Least One")

Understanding "At Least One":
- To find the probability that at least one event occurs out of $n$ trials means finding the probability that 1, 2, 3, …, or $n$ occurrences happen.
- Using the addition rule directly would require calculating individual probabilities for every single non-zero case and summing them all up.
Problem Scenario (500 Females Aged 60):
- Problem: Find the probability that at least one of 500 randomly selected 60-year-old females will die during the course of the year.
- Direct Addition Approach: Would require calculating $P(1 \text{ dies}) + P(2 \text{ die}) + P(3 \text{ die}) + \text{\dots} + P(500 \text{ die})$ , requiring 500 separate calculations.
- Complement Approach: Use the complement rule. The complement of "at least one dies" is "zero die" (which means "all 500 survive").
- Formula: $P(\text{At least one dies}) = 1 - P(\text{None die}) = 1 - P(\text{All 500 survive})$
- Calculation: $P(\text{All 500 survive}) = (0.99186)^{500}$ P(\text{At least one dies}) = 1 - (0.99186)^{500} \text{\approx} 0.9832
- Interpretation: While the probability of all 500 individuals surviving is extremely small, the probability that at least one individual dies among a group of 500 is very high ( $98.32\text{\%}$ or $0.9832$ ).

Summary of Probability Rules

Bounds of Probability: For any event $E$ , the probability must be a real number between 0 and 1 inclusive: 0 \text{\le} P(E) \text{\le} 1
Sum of Outcomes in Sample Space: The sum of the probabilities of all outcomes in a sample space must equal 1: \text{\sum} P(e_i) = 1
Addition Rules:
- For Disjoint Events: $P(E \text{ or } F) = P(E) + P(F)$
- General Addition Rule (Any Events): $P(E \text{ or } F) = P(E) + P(F) - P(E \text{ and } F)$
Complement Rules:
- $P(E) = 1 - P(E^c)$
- $P(E^c) = 1 - P(E)$
- $P(E) + P(E^c) = 1$
Multiplication Rule for Independent Events:
- $P(E \text{ and } F) = P(E) \times P(F)$

Conditional Probability

Definition & Notation:
- The symbol $P(F \mid E)$ uses a vertical pipe symbol ( $\mid$ ) and is read as: "the probability that event $F$ will happen given that event $E$ has already happened."
- The condition following the vertical pipe ( $E$ ) represents the known or given event.
Conditional Probability Formulas:
- Using Probabilities: $P(F \mid E) = \frac{P(E \text{ and } F)}{P(E)}$
- Using Counts/Outcomes (Contingency Table Data): $P(F \mid E) = \frac{N(E \text{ and } F)}{N(E)}$
- Selection between count formula or probability formula depends on which form of data is provided in the problem statement.
Contingency Table Restructure Logic:
- A conditional probability statement narrows the scope of interest by eliminating rows or columns from a contingency table that do not meet the "given" condition.
Identifying Given vs. Target Events in Word Problems:
- Look for modifying phrases (e.g., "who lives in the East").
- Removing the modifier leaves a functional base sentence. The base target question is the event probability being calculated ( $F$ ); the modifier is the given condition ( $E$ ).
Contingency Table Examples (Survey Sample size = 1017 adults):
- Example 1 (Belief in God given region is East):
  - Question: What is the probability that a randomly selected adult American who lives in the East believes in God?
  - Modifier / Given ( $E$ ): Lives in the East.
  - Target ( $F$ ): Believes in God.
  - Count Data: $N(\text{East and Believes in God}) = 204$ . Total residents living in East N(\text{East}) = 204 + 36 + 15 = 255$.\n * Calculation:\n P( ext{Believes in God} \mid ext{Lives in East}) = \frac{204}{255} = 0.80 \text{ (or } 80\% ext{)}\n * *Example 2 (Living in East given Belief in God - Reversed Order)*:\n * Question: What is the probability that a randomly selected adult American lives in the East given that they believe in God?\n * Modifier / Given (E): Believes in God.\n * Target (F): Lives in the East.\n * Denominator change: Sum of all respondents who believe in God across all regions.\n * Calculation:\n P( ext{Lives in East} \mid ext{Believes in God}) = \frac{204}{N( ext{Believes in God})} \text{\approx} 0.26 \text{ (or } 26\% ext{)}\n * *Conclusion on Order*: P(F \mid E) eq P(E \mid F) $. Changing the order alters the given context and yields completely distinct values ($ 80\% $vs$ 26\%).\n\n* **Murder Victims Example (Using Percentages/Probabilities)**:\n * *Data Given*: 16.6\% $($ 0.166 $) of all murder victims were 20-to-24-year-old males.$ 19.1\% $($ 0.191) of all murder victims were 20 to 24 years old.\n * *Problem*: What is the probability that a randomly selected murder victim in 2005 was male given that the victim is 20 to 24 years old?\n * *Rule Note*: Always convert percentages to decimals prior to inserting values into formulas.\n * *Calculation*:\n P( ext{Male} \mid 20\text{--}24) = \frac{P(20\text{--}24 \text{ and Male})}{P(20\text{--}24)} = \frac{0.166}{0.191} \text{\approx} 0.869 \text{ (or } 86.9\% ext{)}\n\n# General Multiplication Rule\n\n* **Formula Derivation**:\n * The General Multiplication Rule is derived directly from the conditional probability formula by algebraically solving for P(E ext{ and } F).\n * *Formulation 1*:\n P(E ext{ and } F) = P(E) \times P(F \mid E)\n * *Formulation 2*:\n P(E ext{ and } F) = P(F) \times P(E \mid F)\n\n* **Variable Swap Strategy**:\n * If variable designations E $and$ F $yield conditional components not supplied in a problem, swap the designations of$ E $and$ F to align with available data.\n * *Verification with Murder Victims Data*:\n P(20\text{--}24 \text{ and Male}) = P(20\text{--}24) \times P( ext{Male} \mid 20\text{--}24) = 0.191 \times 0.869 = 0.166\n\n* **Universal Applicability (Independent vs. Dependent Events)**:\n * The General Multiplication Rule works for **all** events, regardless of whether they are dependent or independent.\n * If events E $and$ F $are independent, then$ P(F \mid E) = P(F) $. Substituting$ P(F) $back into the general rule yields the specific multiplication rule for independent events:$ P(E ext{ and } F) = P(E) imes P(F)$$.

Practical Application Advice and Administrative Information

Homework Precision & Rounding Rules:
- Round numerical outputs to 3 or 4 decimal places as required by individual exercise specifications.
Course Administrative Details:
- Course Registration Number (CRN): 40986.
- Session Date: 23rd.
Conceptual Focus Guidelines:
- Focus heavily on conceptual understanding ("why" a formula is used) rather than simply performing manual calculations.
- Master foundational distinctions between independent vs. dependent events, and disjoint vs. non-disjoint events.