๐ Master Reference ยท Mathematics
A complete study companion โ from sample spaces and classical rules through Bayes' theorem and probability distributions, with worked examples throughout.
Chapter 01
Probability quantifies uncertainty โ it measures how likely an event is to occur on a scale from 0 (impossible) to 1 (certain).
Derived from logical reasoning about equally likely outcomes โ no experimentation required.
P(E) = n(E) / n(S)
Computed from observed data. As trials increase, empirical probability converges toward theoretical probability.
P(E) = freq(E) / total trials
Chapter 02
Chapter 03
P(Aแถ) = 1 โ P(A)
P(S) = 1 ยท P(โ
) = 0
P(AโชB) = P(A) + P(B)
if mutually exclusive
P(AโชB) = P(A) + P(B)
โ P(AโฉB)
if non-exclusive
P(AโฉB) = P(A)ยทP(B)
if independent
P(AโฉB) = P(A)ยทP(B|A)
if dependent
Data visualizations bridge the gap between abstract formulas and intuitive understanding. Probability distributions, sample spaces, and event intersections all become clear when represented graphically โ whether through Venn diagrams, histograms, or scatter plots.
Chapter 04
| Experiment | Sample Space | Example Event | Probability |
|---|---|---|---|
| ๐ฒ Roll a die | {1, 2, 3, 4, 5, 6} | Roll a 4 | P(4) = 1/6 โ 0.167 |
| ๐ Draw a card | 52 cards, 4 suits | Draw a heart | P(โฅ) = 13/52 = 1/4 |
| ๐ช Flip 2 coins | {HH, HT, TH, TT} | Both tails (TT) | P(TT) = 1/4 = 0.25 |
| ๐ญ Factory defect | Machine A + B | Defective item | P(D) = 0.032 |
Chapter 05
The probability of B given that A has already occurred:
P(B | A) = P(A โฉ B) / P(A)
In a class of 40 students, 25 passed both math and physics. The probability that a student passed physics given they passed math = 25/40 = 0.625
When event B can occur through several mutually exclusive pathways Aโ, Aโ, โฆ, Aโ:
P(B) = ฮฃ P(B | Aแตข) ยท P(Aแตข)
Machine A produces 60% of parts (2% defective). Machine B produces 40% (5% defective).
P(Defective) = (0.02 ร 0.6) + (0.05 ร 0.4) = 0.012 + 0.020 = 0.032
Bayes' theorem is the engine of rational inference. A tree diagram reveals how prior probabilities combine with new evidence to produce updated (posterior) probabilities โ the foundation of machine learning, medical diagnostics, and spam filters.
Chapter 06
Reverses conditional probability โ lets you update the probability of a cause given that you've observed an effect.
P(A | B) = [ P(B | A) ยท P(A) ] / P(B)
20% of emails are spam. 90% of spam contains "free". 10% of legitimate email also contains "free".
โ P(spam | "free") = (0.9 ร 0.2) / 0.22 โ 0.818
Chapter 07
Takes a countable number of values. Each value has a specific probability.
Takes any value in a range. Probabilities are defined over intervals (areas under a curve).
| Measure | Description | Formula |
|---|---|---|
| Mean (ฮผ) | Arithmetic average | ฮผ = ฮฃxแตข / n |
| Weighted Mean | Average with weights | ฮผ = ฮฃwแตขxแตข / ฮฃwแตข |
| Median | Middle value when sorted | middle of sorted list |
| Mode | Most frequent value | max frequency |
| Variance (ฯยฒ) | Average squared deviation | ฯยฒ = ฮฃ(xแตขโฮผ)ยฒ / n |
| Std. Dev. (ฯ) | Square root of variance | ฯ = โฯยฒ |
Chapter 08
A probability distribution describes how probability is spread across all possible values of a random variable.
Counts successes in n independent trials, each with probability p.
P(X=k) = C(n,k) ยท pแต ยท (1โp)โฟโปแต
All outcomes are equally likely โ maximum entropy distribution.
P(X = xแตข) = 1/n (discrete)
f(x) = 1/(bโa) (continuous)
The most important distribution in all of statistics. Symmetric around the mean, with probabilities concentrated near the center and tailing off symmetrically. The empirical rule lets you quickly estimate how much data falls within 1, 2, or 3 standard deviations.
f(x) = (1 / ฯโ2ฯ) ยท e^(โ(xโฮผ)ยฒ/2ฯยฒ)
Conclusion
Pro Tip: The best way to internalize probability is through daily problem-solving. Start with classical problems (dice, cards, coins), then move to real-world applications like quality control, medical testing, and machine learning classification โ where Bayes' theorem and distributions dominate.