**Table of Contents**

- Probability Theory
- Seven Problems
- Conditional Probability
- Probability Rules
- Probability Tools
- Bayes Theorem
- Random Variables and Probability Distributions
- Law of Large Numbers
- Central Limit Theorem
- Mendel’s Theory of Heredity
- Epistemic Probability
- Probability Arguments

##### Probability Theory

*Probability Theory*is the formal theory of probability- Developed in 1654 by French mathematicians Pierre de Fermat and Blaise Pascal,
*Probability Theory*is used in physics, genetics, statistics, the social sciences, decision theory, investment analysis, actuarial forecasting, risk assessment, and blocking spam. - These pages set forth basic probability theory by solving seven problems of increasing difficulty. The last is the infamous Monty Hall problem, gotten wrong by hundreds of PhDs.

##### Seven Problems

**Aces and Hearts**- What’s the probability of randomly picking the ace of hearts from a standard deck of 52 cards?
- Also, what are the probabilities of selecting an ace, a heart, an ace or a heart, neither ace nor heart, a red or black card, a black heart?
- View Aces and Hearts

**Flipping Coins**- A normal coin is randomly flipped three times in a row; what’s the probability of three heads?
- Three coins are tossed simultaneously; what’s the probability of three heads?
- View Flipping Coins

**With and Without Replacement**- Three cards are randomly selected from a standard deck. What’s the probability they’re all hearts given that the cards are selected:
- Consecutively with replacement
- Consecutively without replacement
- Simultaneously

- View With and Without Replacement

- Three cards are randomly selected from a standard deck. What’s the probability they’re all hearts given that the cards are selected:
**Sixes**- Three dice are tossed. What’s the probability of rolling:
- Three sixes
- At least one six
- Exactly one six

- View Sixes

- Three dice are tossed. What’s the probability of rolling:
**Cell Phones and Day Packs**- Among 1,000 students 400 have cell phones (C), 300 daypacks (D), and 350 neither. What are the following probabilities for a randomly selected student: P(C&D), P(~C&~D), P(C), P(D), P(CvD), P(~C), P(D|C), P(C|D).
- View Cell Phones and Day Packs

**Random Drug Test**- You’re given a random drug test that’s 95 percent reliable, meaning 95 percent of drug users test positive and 95 percent of non-drug users test negative. Assume five percent of the population takes drugs. You test positive. What’s the probability you use drugs?
- View Random Drug Test

**Monty Hall**- You’re a contestant on a game show and asked to choose one of three closed doors. Behind one door is the car of your dreams; behind the others, goats. You pick a door, Door 1, say, and the host, knowing what’s behind each door, opens one of the other doors, Door 3 for example, revealing a goat. (He always opens a door with a goat.) He then gives you the option of changing your selection to Door 2. Do you have a better chance of winning if you switch; or are the odds the same?
- View Monty Hall

##### Conditional Probability

*Conditional probability*is the probability of something**given that**something else is the case.- Notation
- “P(….. | —–) = n” means that the probability that ….. given that —– equals n.

- The probability of randomly drawing a King from a standard deck of 52 cards is 4/52 = 1/13
- That is,
**P(King) = 1/13**

- That is,
- But the probability of drawing a King given that the card is a Face Card is 1/3, since there are 4 Kings and 12 Face Cards.
- That is,
**P(King | Face Card) = 1/3**, where “|” means “given that.”

- That is,
- The formal definition of
*conditional probability*is**P(A|B) = P(A&B) / P(B),**where P(B) > 0

- Thus,
**P(King | Face Card) = P(King & Face Card) / P(Face Card)**= (4/52) / (12/52) = 4/12 = 1/3. **P(A|B)**is not**P(B|A)****P(King | Face Card) = 1/3**- But
**P(Face Card | King) = 1**

##### Probability Rules

**Probability Tools**

- Calculators
- Natural Frequencies
- Possibility Trees
- Possibility Tables
- Probability Rules
- All Problems

- Probability Trees
- Probability Tables
- Venn Diagrams

**Bayes Theorem**

The core idea of Bayes Theorem is that the probability of a hypothesis given the evidence is, in part, a function of the probability of the evidence given the hypothesis.

View Bayes Theorem

**Random Variables and Probability Distributions**

View Random Variables and Probability Distributions

**Law of Large Numbers**

View Law of Large Numbers

**Central Limit Theorem**

**Mendel’s Theory of Heredity**

Mendel’s Theory uses probability theory to make predictions

View Mendel’s Theory of Heredity

**Epistemic Probability**

The *epistemic probability* of a proposition is how reasonable it is to believe relative to a body of knowledge.