Probability and Statistics

Subject: Mathematics
Topic: 9
Cambridge Code: 0580

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Probability Fundamentals

Basic Concepts

Experiment - Process producing outcomes Sample space (S) - All possible outcomes Event (A) - Subset of sample space Probability - Likelihood of event, 0 ≤ P(A) ≤ 1

Probability Definition

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

Assumes equally likely outcomes

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Probability Rules

Addition Rule

For any events A and B: $P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

Mutually exclusive (A and B cannot occur together): $P(A ∪ B) = P(A) + P(B)$

Multiplication Rule

For independent events: $P(A ∩ B) = P(A) \cdot P(B)$

For dependent events: $P(A ∩ B) = P(A) \cdot P(B|A)$

Complement Rule

$P(A^c) = 1 - P(A)$

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Conditional Probability

Conditional probability - Probability given another event occurred

$P(A|B) = \frac{P(A ∩ B)}{P(B)}, \quad P(B) > 0$

Bayes' Theorem

$P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}$

Generalized: $P(A_i|B) = \frac{P(B|A_i) \cdot P(A_i)}{\sum_j P(B|A_j) \cdot P(A_j)}$

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Distributions

Discrete Distributions

Binomial Distribution - n independent trials, probability p

$P(X = r) = \binom{n}{r}p^r(1-p)^{n-r}, \quad r = 0, 1, ..., n$

Mean: $E(X) = np$ Variance: $\text{Var}(X) = np(1-p)$

Poisson Distribution - Rare events in fixed time/space

$P(X = r) = \frac{e^{-λ}λ^r}{r!}, \quad r = 0, 1, 2, ...$

Mean: $E(X) = λ$ Variance: $\text{Var}(X) = λ$

Continuous Distributions

Normal Distribution - Bell curve, symmetric

$f(x) = \frac{1}{σ\sqrt{2π}}e^{-(x-μ)^2/(2σ^2)}$

Mean: μ Standard deviation: σ

Standardization: $Z = \frac{X - μ}{σ}$ follows standard normal (μ = 0, σ = 1)

Central Limit Theorem

Distribution of sample means approaches normal as n increases, regardless of parent distribution

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Statistics

Measures of Central Tendency

Mean (average): $\bar{x} = \frac{\sum x_i}{n}$

Median - Middle value when ordered

Mode - Most frequent value

Weighted mean: $\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Measures of Spread

Range - Maximum - Minimum

Variance: $s^2 = \frac{\sum (x_i - \bar{x})^2}{n} \text{ (population)}$ $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \text{ (sample)}$

Standard deviation: $s = \sqrt{\text{Variance}}$

Interquartile range (IQR) - $Q_3 - Q_1$

Standardization

$Z = \frac{X - \bar{X}}{s}$

Allows comparison across scales

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Sampling

Sample vs Population

Population - Entire group of interest Sample - Subset used for analysis Bias - Systematic error favoring certain values Random sampling - Every element equally likely

Sampling Methods

Simple random - Every possible sample equally likely Stratified - Divide into strata, sample from each Systematic - Select every kth element Cluster - Divide into clusters, sample clusters

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Hypothesis Testing

Null and Alternative Hypotheses

Null hypothesis $H_0$ - Status quo claim Alternative hypothesis $H_1$ - Research claim

• One-tailed or two-tailed

Significance Level

α - Probability of Type I error

• α = 0.05 (most common)
• α = 0.01 (more stringent)

Type I and Type II Errors

	H₀ True	H₀ False
Reject H₀	Type I error	Correct
Fail to reject H₀	Correct	Type II error

p-value

p-value - Probability of observing data if H₀ true

Decision:

• p < α: Reject H₀
• p ≥ α: Fail to reject H₀

Test Statistics

t-test - Compare means, small samples z-test - Compare means, large samples χ² test - Goodness of fit, independence F-test - Variance comparison

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Correlation and Regression

Correlation Coefficient

Pearson's r - Measures linear relationship (-1 to 1)

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{s_x s_y}$

• r = 1: Perfect positive
• r = 0: No linear relationship
• r = -1: Perfect negative

Linear Regression

Least squares line: $\hat{y} = a + bx$

$b = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} = r\frac{s_y}{s_x}$

$a = \bar{y} - b\bar{x}$

Coefficient of determination: $R^2 = r^2$ - Proportion of variance explained

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Data Visualization

Distributions

Frequency histogram - Shows distribution shape Stem-and-leaf - Shows individual values Box plot - Shows quartiles and outliers

Relationships

Scatter plot - Shows correlation Line graph - Shows trends over time

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Key Points

1. Probability: 0 ≤ P(A) ≤ 1
2. Addition rule for unions
3. Multiplication for intersections
4. Conditional probability changes with information
5. Binomial for discrete, count data
6. Normal for continuous data
7. Central Limit Theorem for sampling
8. Mean and standard deviation describe distributions
9. Hypothesis testing uses significance level
10. Correlation measures relationship strength

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Practice Questions

1. Calculate probabilities using rules
2. Apply Bayes' theorem
3. Find binomial probabilities
4. Use normal distribution tables
5. Calculate sample statistics
6. Conduct hypothesis tests
7. Find confidence intervals
8. Calculate correlation
9. Fit regression lines
10. Interpret results

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Revision Tips

• Know probability rules thoroughly
• Understand normal distribution properties
• Practice hypothesis testing steps
• Understand Type I and Type II errors
• Learn when to use each test
• Interpret correlation correctly
• Practice with real data
• Understand limitations
• Know when normal approximation applies