The Study of Randomness

Chapter 4: Probability: The Study of Randomness

4.1 Randomness

The Language of Probability

Chance Behavior: Describes unpredictable outcomes in the short term but reveals a regular and predictable pattern over the long term.
Random Phenomenon: Defined as a situation where individual outcomes are uncertain, yet there is a consistent distribution of outcomes when repeated multiple times.
Probability of Outcomes: The probability of any outcome is expressed as the ratio of how often the outcome occurs over many repetitions.

Thinking about Randomness

Single Trial vs. Many Trials: A single coin toss result is random, but outcomes over multiple tosses can be predicted if trials are independent (i.e., the outcome of one flip does not affect another).

Uses of Probability

Origins: Originating from 17th-century games of chance.
18th & 19th Century Advances: Advances from astronomy and surveying measurements led to significant developments in probability theory due to distributions arising from random sampling.
Modern Applications: Modern probability is utilized in diverse fields including:
- Traffic flows
- Genetic compositions of populations
- Subatomic particle energy states
- Spread of diseases or social media tweets
- Returns on risky investments

4.2 Probability Models

Key Concepts

Descriptions of chance behavior consist of:
- A list of possible outcomes
- A probability assigned to each outcome.
Sample Space (S): The comprehensive set of all potential outcomes.
Event: Defined as either a specific outcome or a set of outcomes, essentially a subset of the sample space.
Probability Model: Comprised of the sample space S and a probability for each outcome.

Example: Rolling Two Dice

For rolling two fair, six-sided dice:
- Sample Space: 36 outcomes (since each die has 6 faces, and 6 × 6 = 36).
- Probability: Each outcome possesses an equal probability of $\frac{1}{36}$ .

Probability Rules

Probability Range: Any probability is a number between 0 and 1.
Total Probability: The sum of all possible outcomes must equal 1.
Disjoint Outcomes: If two events have no outcomes in common, the probability of either occurring is the sum of their individual probabilities.
Complementary Events: The probability that an event does not occur is given by $P(A^C) = 1 - P(A)$ .

Example: Vehicle Color Preferences

A probability model can be represented as:
- Color: White, Black, Silver, Gray, Red, Blue, Brown, Other
- Probability: 0.24, 0.19, 0.16, 0.15, 0.10, 0.07, 0.05, 0.04.
Legitimacy Check:
- Ensuring each probability lies between 0 and 1, and their total equals 1.
Calculations:
- P(Black or Silver) = $P(Black) + P(Silver) = 0.19 + 0.16 = 0.35$ .
- P(Not Blue) = $1 - P(Blue) = 1 - 0.07 = 0.93$ .

Venn Diagrams

Venn Diagram: Utilized to visually represent relations among various events within a sample space S.
- Disjoint Events: Events without any common outcomes.

Assigning Probability: Finite Probability Models

To assign probabilities to events within a finite model:
- Assign individual probabilities to outcomes that sum to 1.
- Probability for an event = sum of probabilities of its constituent outcomes.

Assigning Probability: Equally Likely Outcomes

In many cases, outcomes may be considered equally likely, especially in scenarios like tossing coins or rolling dice.
If a random process has $k$ equally likely outcomes, the probability for any individual outcome is given by: $P = \frac{1}{k}$ .

Independence and the Multiplication Rule

Independent Events: Two events A and B are independent if the occurrence of one does not affect the probability of the other happening.
Multiplication Rule: If A and B are independent, then:
$P(A ext{ and } B) = P(A) imes P(B).$

Example: Rapid HIV Tests

Scenario: In a clinic testing HIV, with a 2% false-positive rate:
- The probability of a false positive for a single test is $P(Pos) = 0.02$ .
- Probability of no false positives for 50 tests is given by $P(Negative) = 0.98$ , leading to:
  $P(50 ext{ negatives}) = 0.98^{50} \, ext{and} \, P(at least 1 ext{ positive}) = 1 - P(50 \text{ negatives})$ .

4.3 Random Variables

Random Variable Definition

A random variable assigns numerical values tied to the outcomes of a random phenomenon.
Probability Distribution: Illustrates possible values and their corresponding probabilities.

Example: Tossing a Coin

For tossing a fair coin thrice with variable $X$ signifying the number of heads:
- outcomes: TTT, HTT, THT, TTH, HHT, HTH, THH, HHH
- Distribution:
- X = 0
  ightarrow TTT
  ightarrow P = rac{1}{8}
- X = 1
  ightarrow HTT, THT, TTH
  ightarrow P = rac{3}{8}
- X = 2
  ightarrow HHT, HTH, THH
  ightarrow P = rac{3}{8}
- X = 3
  ightarrow HHH
  ightarrow P = rac{1}{8}.

Discrete Random Variables

Definition: Discrete random variables can be explicitly listed and have fixed values.
Probability Requirements:
1. Each probability must be between 0 and 1.
2. The total probability must sum to 1.

Example: College Grade Distribution

Grade Points Distribution: A = 4, B = 3, C = 2, D = 1, F = 0.
Given distributions: 32% A's, 42% B's, etc.
To find: P(X ≥ 3) = $P(X = 3) + P(X = 4) = 0.42 + 0.32 = 0.74$ .

Continuous Random Variables

Continuous random variables represent outcomes founded on measurement.
Definition: Y can take any value within an interval and is illustrated by a density curve.
Probability Calculations: Only ranges yield positive probabilities rather than individual values.

Probability Models for Continuous Variables

Selecting a number between 0 and 1 with probabilities defined as areas beneath density curves.
Probability Example: P(X ≤ 0.5 or X > 0.8) = P(X ≤ 0.5) + P(X > 0.8) = 0.5 + 0.2 = 0.7.

Normal Probability Models

There are specific distribution curves like the Normal curve for assigning probabilities.
Standard Normalization: Transforming any Normal curve N(m, s) to standard Normal curve N(0, 1) via z-scores.
Example: women's height with mean 64.5, standard deviation 2.5
- Determine P(68 < X < 70) using z-scores:
  P(1.4 < Z < 2.2) = P(Z < 2.2) – P(Z < 1.4) = 0.986 – 0.919 = 0.067.

4.4 Means and Variances of Random Variables

Mean of a Random Variable

The mean of discrete random variable X is computed by weighting possible outcomes.
$ext{Mean} = ext{Expected Value} = ext{sum of} X_i imes P(X_i)$ .

Example: Apgar Scores

Example probability distribution for Apgar scores:
- Value distribution: 0 through 10.
- Calculating the mean:
  ext{Mean Apgar score}
  ightarrow 8.128.
- Interpretation: Indicates the long-term average for Apgar scores across many newborns.

Statistical Estimation

Key challenge: Estimating the unknown population mean $ext{µ}$ through sample means, recognizing variability in sample means.

Law of Large Numbers

As the sample size increases, the sample mean approaches the population mean $ext{µ}$ .
Note: The law of small numbers does not hold; short sequences may misrepresent average behavior expected in the long run.

Rules for Means

Rule 1: $ext{µ}_{a+bX} = a + b ext{µ}_X$ .
Rule 2: $ext{µ}_{X+Y} = ext{µ}_X + ext{µ}_Y$ .
Rule 3: $ext{µ}_{X-Y} = ext{µ}_X - ext{µ}_Y$ .

Variance of a Random Variable

Variance captures spread, similar to quantitative data:
The variance formula:

Example: Apgar Variability

Variance of X = Apgar score is utilized to calculate standard deviation:
$ext{Standard Deviation} = ext{about } 1.4.$

Rules for Variances

Rule 1: $ext{σ}^2_{a+bX} = b^2σ^2_X$ .
Rule 2: $ext{σ}^2_{X+Y} = σ^2_X + σ^2_Y$ .
Rule 3: If X and Y have correlation
ho:
$ext{σ}^2_{X ext{ and } Y} = σ^2_X + σ^2_Y + 2ρσ_Xσ_Y$ .

4.5 General Probability Rules

Key General Rules

$0 ≤ P(A) ≤ 1$ for any event A.
$P(S) = 1$ for the sample space S.
If A and B are disjoint, $P(A or B) = P(A) + P(B)$ .
Complementary Events: $P(A^C) = 1 - P(A)$ .
Independence: $P(A ext{ and } B) = P(A)P(B).$

Venn Diagrams

Visual representations help clarify relationships among events
Diagrams effectively illustrate sample spaces and disjoint/non-disjoint scenarios.

General Addition Rule

Disjoint Events: $P(A ext{ or } B) = P(A) + P(B).$
For events that may share outcomes (non-disjoint):
$P(A ext{ or } B) = P(A) + P(B) – P(A ext{ and } B).$

Conditional Probability

Conditional probability varies depending on additional known events:
- If event A occurs, adjust the probability of event B accordingly:
  $P(B | A) = \frac{P(A ext{ and } B)}{P(A)}.$

General Multiplication Rule

For two events A and B,
$P(A ext{ and } B) = P(A) P(B | A).$

Tree Diagrams

Utilize tree diagrams to model multi-step probability scenarios:
- E.g., Flipping two coins yields: Outcomes: HH, HT, TH, TT with respective probabilities of 1/4 for two heads.

Bayes's Rule

Applies conditional probabilities within a segmented probability space:
- If A1, A2, …, Ak represent mutually exclusive events in a total probability space, event C can be expressed using:
  $P(A_i | C) = \frac{P(C | A_i)P(A_i)}{P(C)}$ .

Bayes's Rule Example

In a scenario involving mammography testing for breast cancer, calculate the probability of disease post-positive test results, using prior knowledge of disease incidence and test accuracy metrics.

Independence Again

Events A and B are independent if the occurrence of one does not alter the probability of the other, represented by:
- $P(A | B) = P(A)$ for events both taking positive probabilities.

Review Summary

Section Recaps:
- 4.1 Randomness
- 4.2 Probability Models
- 4.3 Random Variables
- 4.4 Means and Variances
- 4.5 General Probability Rules