Probability
Basic Probability Concepts
Section titled “Basic Probability Concepts”Sample Space and Events
Section titled “Sample Space and Events”- Sample space (): The set of all possible outcomes of an experiment
- Event: Any subset of the sample space
- Probability: A number between 0 and 1 that measures the likelihood of an event
Probability Rules
Section titled “Probability Rules”- Legitimate probability values: for any event
- Sum of probabilities: (the probability of the entire sample space is 1)
- Complement rule: , where is the complement of
Addition Rule
Section titled “Addition Rule”For any two events and :
P(A \cup B) = P(A) + P(B) - P(A \cap B)
If and are mutually exclusive (disjoint, cannot both occur): P(A \cup B) = P(A) + P(B)
Multiplication Rule
Section titled “Multiplication Rule”P(A \cap B) = P(A) \cdot P(B | A)
If and are independent: $$P(A \cap B) = P(A) \cdot P(B)
Conditional Probability
Section titled “Conditional Probability”P(B | A) = \frac{P(A \cap B)}{P(A)}
The probability of given that has occurred. Note that must be greater than 0.
Independence
Section titled “Independence”Events and are independent if the occurrence of one does not affect the probability of the other:
P(B | A) = P(B) \quad \text{and} \quad P(A | B) = P(A)
Equivalently:
Important: Independence is not the same as mutually exclusive. In fact, if two events are both mutually exclusive and both have non-zero probability, they cannot be independent.
Checking Independence
Section titled “Checking Independence”To check independence on the AP exam:
- Calculate and separately
- Calculate
- Check whether
Alternatively, check whether .
Disjoint vs Independent
Section titled “Disjoint vs Independent”| Property | Disjoint (Mutually Exclusive) | Independent |
|---|---|---|
| Definition | Cannot occur together | Occurrence of one does not affect the other |
Bayes” Theorem
Section titled “Bayes” Theorem”P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B | A) \cdot P(A) + P(B | A^c) \cdot P(A^c)}
Used to find the probability of a “cause” given an observed “effect.” Useful for medical testing problems.
Two-Way Tables and Probability
Section titled “Two-Way Tables and Probability”Given a two-way table:
- Marginal probability: probability based on a single variable (row or column total / grand total)
- Joint probability: probability of both events occurring (cell / grand total)
- Conditional probability: probability given a condition (cell / row total or cell / column total)
Discrete Random Variables
Section titled “Discrete Random Variables”A random variable assigns a numerical value to each outcome in the sample space.
Probability Distribution
Section titled “Probability Distribution”A table, graph, or formula that gives the probability () for each value () of the random variable .
Requirements:
- for each
Mean (Expected Value)
Section titled “Mean (Expected Value)”\mu_X = E(X) = \sum x_i \cdot p_i
The mean of a random variable is the long-run average of its values over many repetitions.
Variance and Standard Deviation
Section titled “Variance and Standard Deviation”\sigma_X^2 = \sum (x_i - \mu_X)^2 \cdot p_i = \sum x_i^2 \cdot p_i - \mu_X^2
\sigma_X = \sqrt{\sigma_X^2} ### Rules for Means and Variances For random variables $X$ and $Y$, and constants $a$ and $b$: - $\mu_{a + bX} = a + b\mu_X$ - $\mu_{X + Y} = \mu_X + \mu_Y$ (always) - $\sigma_{a + bX}^2 = b^2\sigma_X^2$ - If $X$ and $Y$ are independent: $\sigma_{X + Y}^2 = \sigma_X^2 + \sigma_Y^2$ and $\sigma_{X - Y}^2 = \sigma_X^2 + \sigma_Y^2$ ## Binomial Distributions A binomial setting has four conditions: 1. **Binary**: Each observation has two possible outcomes (success/failure) 2. **Independent**: Observations are independent (or approximately independent if sampling without replacement and the population is at least 10 times the sample) 3. **n trials**: A fixed number of trials $n$ 4. **p probability**: The probability of success $p$ is the same for each trial ### Binomial Probability P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} ### Mean and Standard Deviation \mu = np, \quad \sigma = \sqrt{np(1-p)} ## Geometric Distributions A geometric setting has three conditions: 1. **Binary**: Each trial has two outcomes 2. **Independent**: Trials are independent 3. **p probability**: The probability of success is the same for each trial The random variable $X$ counts the **number of trials until the first success**. ### Geometric Probability P(X = k) = (1-p)^{k-1} \cdot p ### Mean \mu = \frac{1}{p} ## Normal Distributions and the Central Limit Theorem ### Central Limit Theorem (CLT) For a sufficiently large sample size $n$ ($n \geq 30$ is the standard threshold), the sampling distribution of $\bar{x}$ is approximately normal, regardless of the shape of the population distribution. \mu_{\bar{x}} = \mu, \quad \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} The CLT allows us to use normal probability calculations for sample means even when the population is not normally distributed. ### Sampling Distribution of $\hat{p}$ For a sample proportion $\hat{p}$ with population proportion $p$ and sample size $n$: \mu_{\hat{p}} = p, \quad \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} Approximately normal when $np \geq 10$ and $n(1-p) \geq 10$. ## Intuition Probability is the **mathematics of uncertainty**, it quantifies how likely events are and provides the foundation for statistical inference. The key insight is that probability statements are about *long-run frequencies*, not single events. **The addition rule intuition:** $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. The subtraction of the intersection prevents double-counting, if you count all of A and all of B, you've counted the overlap twice. **Conditional probability intuition:** $P(A|B) = P(A \cap B) / P(B)$ restricts the sample space to event B. Think of it as "given that B has occurred, what fraction of those outcomes also have A?" Independence means $P(A|B) = P(A)$, knowing B occurred doesn't change the probability of A. **Bayes' theorem intuition:** Bayes' theorem flips conditional probabilities: if you know $P(B|A)$ and want $P(A|B)$, you need to account for the base rates of both events. This is why medical test results must be interpreted in context, a positive test for a rare disease may still have a low probability of being a true positive. **The Central Limit Theorem:** This is the bridge between probability and inference. No matter what the population distribution looks like, the sampling distribution of the mean becomes approximately normal for large $n$. This is why z-scores and t-scores work, they rely on the CLT. ## Common Pitfalls - Confusing $P(A | B)$ with $P(B | A)$ - Assuming events are independent without justification - Confusing disjoint with independent - Forgetting to check the 10% condition for the binomial approximation - Misapplying the central limit theorem with small sample sizes $ - Confusing the mean of a random variable with its most probable value ## Cross-References - **Exploring Data:** Probability distributions describe the theoretical models behind the data you explore. - **Statistical Inference:** The Central Limit Theorem is the foundation of all inference procedures. - **Sampling:** Random sampling relies on probability to ensure representative samples.