Chapter 4: Probability

Section 4.1: Basic Concepts of Probability

Probability is a foundational concept in statistics, used to quantify uncertainty and make decisions based on data. It is central to hypothesis testing and helps determine whether observed outcomes are due to chance or other factors.

• Event: Any collection of results or outcomes from a procedure.

• Simple Event: An outcome that cannot be broken down further.

• Sample Space: The set of all possible simple events for a procedure.

• Notation: P denotes probability; A, B, C denote specific events; P(A) is the probability of event A.

Example: For a single birth, the sample space is {g, b} (girl, boy). For three births, the event "2 girls and 1 boy" is not a simple event, as it can occur in multiple ways (ggb, gbg, bgg).

Approaches to Probability

• Relative Frequency Approximation: Probability is estimated by the proportion of times an event occurs in repeated trials.

• Classical Approach: Used when outcomes are equally likely.

• Subjective Probability: Based on personal judgment or experience.

Law of Large Numbers: As a procedure is repeated many times, the relative frequency probability approaches the actual probability.

Probability Properties

• means event never happens; means event always happens.

• The complement of event A, denoted or , consists of all outcomes where A does not occur.

Example: In a survey, 366 adults said yes to seeing a ghost, 1637 said no. Probability a randomly selected adult said yes:

Rounding Probabilities: Probabilities are often rounded to three significant digits.

Significant Numbers of Successes

• Significantly High: x successes in n trials is significantly high if

• Significantly Low: x successes in n trials is significantly low if

• The threshold 0.05 is common but not absolute; other values like 0.01 may be used.

Odds

• Actual Odds Against: (expressed as a:b)

• Actual Odds In Favor: (reciprocal of odds against)

• Payoff Odds: Ratio of net profit to amount bet.

Example: A roulette wheel has 38 slots. Probability of winning by betting on an even number:

Example: Betting P(\text{win}) = \frac{1}{38}P(\text{lose}) = \frac{37}{38}$ Net profit if win: $5 \times 35 = $175

Section 4.2: Addition Rule and Multiplication Rule

Compound events involve combinations of two or more simple events. The addition and multiplication rules help calculate probabilities for these events.

Addition Rule

• General Addition Rule: For events A and B:

• Disjoint (Mutually Exclusive) Events: Events that cannot occur together. For disjoint events, , so

• Rule of Complementary Events:

Example Table: High school drivers and risky behaviors:

Applications:

• Probability of selecting a driver who drove while drinking:

• Probability of selecting a driver who did not text while driving:

• Probability of selecting a driver who texted while driving or drove while drinking: Use addition rule.

Multiplication Rule

• General Multiplication Rule:

• Conditional Probability: is the probability of B occurring given A has occurred.

• Independent Events: If A and B are independent, , so

• Dependent Events: If A and B are not independent, use conditional probability.

Sampling:

• With Replacement: Selections are independent.

• Without Replacement: Selections are dependent.

• Approximate Independence: If sample size is no more than 5% of population, treat as independent.

Example: Drug testing 50 adults (45 positive, 5 negative). Probability both selected have positive result:

• With replacement:

• Without replacement:

Example: Probability all three randomly selected employees test positive (population 130,639,273, positive rate 0.042): (approximate independence)

Example: Hard drive failure rate 3.66%. Probability both primary and backup fail within a year:

Additional info: The notes provide practical examples and reinforce the importance of understanding independence, conditional probability, and the distinction between different probability rules. These concepts are foundational for later topics such as hypothesis testing and statistical inference.