Probability, Discrete and Binomial Distributions, and Normal Distributions: Study Guide (Chapters 4–6)

Probability and Basic Concepts

Basic Probability Concepts

Probability quantifies the likelihood of an event, always taking a value between 0 and 1. Understanding probability requires familiarity with outcomes, events, and the sample space.

• Outcome: A single possible result of a random experiment.

• Simple event: An event that cannot be decomposed into simpler outcomes.

• Sample space (S): The set of all possible simple outcomes.

• Event: Any collection (subset) of outcomes from the sample space.

• Probability of event A: Denoted P(A).

Equally likely outcomes:

• $P(A) = \frac{\text{number of favorable outcomes}}{\text{total possible outcomes}}$

• Example: For a fair die, $S = \{1,2,3,4,5,6\}$, $P(5) = 1/6$, $P(\text{even}) = 3/6 = 1/2$.

Three approaches to probability:

• Classical: Assumes equally likely outcomes.

• Relative-frequency: Uses observed frequency over number of trials.

• Subjective: Based on judgment or experience.

Law of Large Numbers: As the number of trials increases, the observed relative frequency approaches the true probability.

Unusual events: Events with probability less than 0.05 (5%) are often considered unusual, but context matters.

Addition and Multiplication Rules; Disjointness and Independence

Probability rules help determine the likelihood of combined events.

• Addition Rule (OR): $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• If A and B are disjoint (mutually exclusive), $P(A \text{ and } B) = 0$.

• Multiplication Rule (AND): $P(A \text{ and } B) = P(A) \times P(B|A)$

• If A and B are independent, $P(B|A) = P(B)$, so $P(A \text{ and } B) = P(A) \times P(B)$.

• Disjoint vs. Independent: Disjoint events cannot occur together; independent events do not affect each other's probabilities.

Replacement: With replacement, probabilities reset after each selection. Without replacement, probabilities change as items are removed. The 5% guideline: If the sample is no more than 5% of the population, selections can often be treated as independent for calculations.

Worked Example (Cards): In a 52-card deck, $P(\text{king or heart}) = 4/52 + 13/52 - 1/52 = 16/52 = 4/13$ (the king of hearts is counted in both groups).

Worked Example (Without Replacement): From 5 red and 3 blue marbles, $P(\text{two reds}) = (5/8) \times (4/7) = 20/56 = 5/14$.

Complements and Conditional Probability

Complements and conditional probability are essential for solving complex probability problems.

• Complement: $P(\text{not } A) = 1 - P(A)$

• "At least one" is often calculated as $1 - P(\text{none})$

• For independent trials with event probability p: $P(\text{at least one in } n \text{ trials}) = 1 - (1-p)^n$

• Conditional Probability: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$, provided $P(A) > 0$

Worked Example (Complement): Seven independent days, each with 5% rain probability: $P(\text{at least one rainy day}) = 1 - (0.95)^7 \approx 0.3017$

Worked Example (Conditional Probability): In a group of 140 part-time students (80 female, 60 male), $P(\text{female} | \text{part-time}) = 80/140 = 4/7 \approx 0.5714$

Common Probability Traps

• "Or" includes either event or both, unless events are disjoint.

• "And" generally requires multiplication.

• Probabilities must be between 0 and 1.

• For complements, rewrite 1 as a common denominator before subtracting fractions.

Discrete Probability Distributions

Random Variables and Distributions

A random variable assigns a numerical value to each outcome of a random process.

• Discrete random variable: Takes countable values (e.g., number of children).

• Continuous random variable: Takes any value in an interval (e.g., time, height).

• Discrete probability distribution: Assigns a probability $P(x)$ to each possible value $x$.

• Requirements: $0 \leq P(x) \leq 1$ for all $x$, and $\sum P(x) = 1$.

Mean (Expected Value): $\mu = \sum [x \times P(x)]$

Variance: $\sigma^2 = \sum [(x - \mu)^2 \times P(x)] = \sum [x^2 \times P(x)] - \mu^2$

Standard Deviation: $\sigma = \sqrt{\sigma^2}$

Expected Value (E): $E = \sum [x \times P(x)]$

• For money problems, x should represent net profit or loss (subtract cost from payout).

Worked Example: $X = 0, 1, 2$ with probabilities $0.2, 0.5, 0.3$:

• Check: $0.2 + 0.5 + 0.3 = 1$

• Mean: $0(0.2) + 1(0.5) + 2(0.3) = 1.1$

• $E(X^2) = 0^2(0.2) + 1^2(0.5) + 2^2(0.3) = 0 + 0.5 + 1.2 = 1.7$

• Variance: $1.7 - (1.1)^2 = 0.49$

• SD: $\sqrt{0.49} = 0.7$

Binomial Distributions

The binomial distribution models the number of successes in a fixed number of independent yes/no trials.

• Requirements:

  1. Fixed number of trials $n$

  2. Independent trials

  3. Exactly two outcomes (success/failure)

  4. Constant probability of success $p$

• Notation: $n$ = trials, $p$ = success probability, $q = 1-p$, $x$ = number of successes

• Probability of exactly x successes: $P(X = x) = C(n, x) p^x q^{n-x}$

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

• Standard deviation: $\sigma = \sqrt{npq}$

Calculator references:

• Exactly x: binompdf(n, p, x)

• At most x: binomcdf(n, p, x)

• Fewer than x: binomcdf(n, p, x-1)

• At least x: $1 - \text{binomcdf}(n, p, x-1)$

• More than x: $1 - \text{binomcdf}(n, p, x)$

• Between a and b inclusive: $\text{binomcdf}(n, p, b) - \text{binomcdf}(n, p, a-1)$

Worked Example: Five trials, $p = 0.2$:

• $P(\text{at least one success}) = 1 - P(0) = 1 - 0.32768 = 0.67232$

• $P(\text{exactly two}) = C(5,2)(0.2)^2(0.8)^3 = 10 \times 0.04 \times 0.512 = 0.2048$

Phrase	Mathematical Expression / Calculator
Exactly 6	binompdf(n, p, 6)
At most 6 / no more than 6	binomcdf(n, p, 6)
Fewer than 6	binomcdf(n, p, 5)
At least 6	1 - binomcdf(n, p, 5)
More than 6	1 - binomcdf(n, p, 6)
Between 3 and 8, inclusive	binomcdf(n, p, 8) - binomcdf(n, p, 2)

Normal Distributions and Sampling

Continuous and Standard Normal Distributions

Continuous probability distributions assign probabilities as areas under a curve. The normal distribution is the most important continuous distribution in statistics.

• Continuous distribution: Probability is area under the density curve; total area = 1.

• Probability at a single point is 0: $P(X < a) = P(X \leq a)$ for continuous X.

• Uniform distribution: Constant density across an interval. $P(a < X < b) = \frac{b - a}{\text{total interval width}}$

• Normal distribution: Symmetric, bell-shaped, with mean $\mu$ and standard deviation $\sigma$.

• Standard normal: $\mu = 0$, $\sigma = 1$

• Empirical Rule: About 68%, 95%, and 99.7% of values fall within 1, 2, and 3 SDs of the mean, respectively.

Calculator functions:

• Find area: normalcdf(lower, upper, $\mu$, $\sigma$)

• Find cutoff from area: invNorm(area, $\mu$, $\sigma$)

Worked Example: $P(-1 < Z < 1) \approx 0.6827$; $P(Z > 1.35) \approx 0.0885$; 90th percentile $z \approx 1.2816$; cutoff with 5% to the right $z \approx 1.6449$.

Real Applications and z-Scores

The z-score standardizes values for comparison and probability calculations.

• Population z-score: $z = \frac{x - \mu}{\sigma}$

• Sample z-score: $z = \frac{x - \bar{x}}{s}$

• To convert back: $x = \mu + z\sigma$

• Positive z: above mean; negative z: below mean.

Worked Example: Exam scores with $\mu = 75$, $\sigma = 8$, 1.5 SDs below average: $x = 75 - 1.5 \times 8 = 63$

Calculator usage: Given x, find probability → normalcdf; given percentile, find x → invNorm.

Sampling Distributions and Unbiased Estimators

A sampling distribution describes the distribution of a statistic (like the sample mean) over all possible samples from a population.

• Sampling distribution of $\bar{x}$: Distribution of sample means, not individual values.

• Unbiased estimator: Statistic whose sampling distribution is centered on the population parameter.

• Sample mean $\bar{x}$ is unbiased for $\mu$; sample variance $s^2$ is unbiased for $\sigma^2$ (under random sampling).

Worked Example: Drawing two times with replacement from balls numbered 1, 2, 3:

• Possible sample means: 1, 1.5, 2, 2.5, 3 with frequencies 1, 2, 3, 2, 1 (out of 9 total).

• Mean of sampling distribution: 2 (same as population mean).

Central Limit Theorem (CLT)

The CLT is fundamental for making inferences about population means using sample means.

• For independent random samples from a population with mean $\mu$ and SD $\sigma$:

• Sampling distribution of $\bar{x}$ has mean $\mu_{\bar{x}} = \mu$ and standard error $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• If the population is normal, $\bar{x}$ is normal for any $n$.

• If the population is not normal, $\bar{x}$ is approximately normal for large $n$ (rule of thumb: $n \geq 30$).

• Larger samples have smaller standard error; sample means cluster more tightly around $\mu$.

Worked Example: GPA population $\mu = 3.02$, $\sigma = 0.29$, $n = 25$:

• Standard error: $0.29/\sqrt{25} = 0.058$

• $P(\bar{x} > 3.20)$: $z = (3.20 - 3.02)/0.058 \approx 3.10$, probability $\approx 0.0010$

• For a single student: $z = 0.18/0.29 \approx 0.62$ (less extreme)

CLT Workflow:

1. Determine if the question concerns an individual or a sample mean.

2. Verify normal model (population normal or large $n$).

3. Use mean $\mu$ and SD $\sigma/\sqrt{n}$ for $\bar{x}$.

4. Compute area with normalcdf, or cutoff with invNorm.

Practice and Application

Practice problems and their solutions reinforce the above concepts. Key skills include identifying the correct probability rule, distinguishing between discrete and continuous variables, validating probability distributions, calculating expected values, and applying the binomial and normal models appropriately.

Summary Table: Key Probability Rules

Situation	Rule/Formula
Addition (OR)	$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$
Multiplication (AND, independent)	$P(A \text{ and } B) = P(A) \times P(B)$
Multiplication (AND, dependent)	$P(A \text{ and } B) = P(A) \times P(B|A)$
Complement	$P(\text{not } A) = 1 - P(A)$
Conditional Probability	$P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$
Binomial Probability	$P(X = x) = C(n, x) p^x q^{n-x}$
Normal z-score	$z = \frac{x - \mu}{\sigma}$
Sample Mean SE	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Final Self-Check Questions

• Am I finding an exact count, a cumulative count, an area, or a cutoff?

• Did I translate "less than" versus "at most" correctly?

• Does replacement change the denominator?

• For conditional probability, did I use the correct denominator?

• For a sample mean, did I divide $\sigma$ by $\sqrt{n}$?

• Is my answer between 0 and 1, and did I round only at the end?

Additional info: These notes cover Sections 4.1–4.3, 5.1–5.2, and 6.1–6.4, including probability rules, discrete and binomial distributions, normal distributions, sampling distributions, and the central limit theorem. Practice problems and answer keys are included for self-assessment.