aka “set theory”

\begin{align} A \cup B &&& \text{union (A or B)}\\ A \cap B &&& \text{intersection (A and B)}\\ A \setminus B &&& \text{B substracted from A (A minus B)}, also \\ &&& \text{relative complement of B in A (A and not B)}\\ \\ A \subset B &&& \text{A is a proper subset of B}\\ A \subseteq B &&& \text{A is a subset of B} \\ A \nsubseteq B &&& \text{A is not a subset of B} \\ A \supset B &&& \text{A is a proper superset of B} \\ A \supseteq B &&& \text{A is a superset of B} \\ A \nsupseteq B &&& \text{A is not a superset of B} \\ \\ x \in A &&& \text{x is in A} \\ x \notin A &&& \text{x is not in A} \\ A \owns x &&& \text{A owns x} \\ \end{align}

A subset can be equal to its super. A proper subset cannot.

Note. Salman Khan uses the term “strict” subset as synonymous with “proper” subset, and he uses a different symbol for it.

Subsets exist within a universe. The absolute complement of a set, is everything in the universe not in the set. When comparing two sets, the relative complement means relative to the other set.

\begin{align} P\left ( A \right ) &&& \text{probability of A}\\ \end{align}

probability: number between 0 and 1

ratio: 1 to 4, 1:4, 1 out of four

odds: 1 to 3

fraction: 1/4

decimal: .25

percentage: 25

1 = certainty

0 = impossible

ratio of one outcome to another: 1 to 3 ratio of one outcome to the total possibilities: 1 out of 4

probability .25

odds .25 / .75

six-sided die probability of rolling a 3: 1/6 probability of not rolling a 3: 5/6 odds of rolling a 3: 1/6 / 5/6 or 1/5

Probability

Odds

Ratio

sample space - the set of all the possible outcomes

subsets of sample space - the subset of those outcomes that satisfy a condition

theoretical probability - calculated, like a coin or a die

experimental probability - derived from data

trial - one flip of the coin, one roll of the die

law of large numbers - as the number of trials increases, the experimental probability tends to match the theoretical probability

simulations using random numbers can be used to calculate experimental probability

statistical significance - if the probability of an experimental outcome happening by chance is greater than 5%, then it is significant

Use a venn diagram to represent union vs intersection of two subsets within the sample space.

A compound sample space has multiple variables. Ways that an event can vary.

A sample space of multiple independent variables can be represented graphically.

• venn diagram, for a small number of variables
• tree diagram, for a small number of variables
• grid, for two variables
• table, for two or three variables

To calculate the probability of two or more independent events, multiply the individual probabilities together.

$P \left ( n \: \text{throws in a row} \right ) = P \left (\text{1 throw} \right ) ^n$

3 coin flips

3 throws of the die

people in chairs permutations combinations

red balls and blue balls

How many total possible outcomes of 8 coin flips? = $2^{8}$

\begin{align} _{n}P_{r} &= \frac{n!}{(n-r)!} \\ _{n}C_{r} &= \frac{n!}{(n-r)!r!} \\ \end{align}

independent events - with replacement

dependent events - without replacement, first event changes the sample space

\begin{align} P(A) &&& \text{probability of A} \\ P(B) &&& \text{probability of B} \\ P(B | A) &&& \text{probability of B given A} \\ \end{align}

Applies to dependent events.

If:
\begin{align} P(A) &= \text{probability of A} \\ P(B) &= \text{probability of B} \\ P(B \cap A) &= \text{intersection of B and A} \\ P(B \mid A) &= \text{probability of B given A} \\ \end{align}
Then: \begin{align} P(B \mid A) &= \frac{P(B \cap A)} {P(B)} \\ \end{align}
And: \begin{align} P(A \mid B) &= \frac{P(A \cap B)} {P(A)} \\ \end{align}

Named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event.

For example, if the risk of developing health problems is known to increase with age, Bayes' theorem allows the risk to an individual of a known age to be assessed more accurately (by conditioning it on their age) than simply assuming that the individual is typical of the population as a whole.

Wikipedia: Bayes' Theorem