Probability of $A$, $\mathrm{P}\left( A \right)=$ Probability of $A'$, $\mathrm{P}\left( A' \right)=$ Probability of $B$, $\mathrm{P}\left( B \right)=$ Probability of $B'$, $\mathrm{P}\left( B' \right)=$ Probability of $A\cap B$, $\mathrm{P}\left( A\cap B \right)=$ Probability of $A'\cap B'$, $\mathrm{P}\left( A'\cap B' \right)=$ Probability of $B$ given that $A$ happens, $\mathrm{P}\left( B|A \right)=$ Probability of $B'$ given that $A$ happens, $\mathrm{P}\left( B'|A \right)=$ Probability of $B$ given that $A'$ happens, $\mathrm{P}\left( B|A' \right)=$ Probability of $B'$ given that $A'$ happens, $\mathrm{P}\left( B'|A' \right)=$ $A$$A'$ $B$$w$$x$ $B'$$y$$z$

Tables of Outcomes, Venn Diagrams and Probability Tables

Four commonly used probability diagrams are tables of outcomes, Venn diagrams, probability tables and tree diagrams.

Table of outcomes

A table of outcomes tabulates all possible outcomes when the sample space is finite and not large.

Rows and columns list the possible outcomes of two random experiments.
Each cell is the outcome obtained when its row and column outcomes occur together.
If the sample space is very large, use permutations and combinations where necessary.

For a coin toss followed by a six-sided die roll:

	1	2	3	4	5	6
Tail	(T,1)	(T,2)	(T,3)	(T,4)	(T,5)	(T,6)
Head	(H,1)	(H,2)	(H,3)	(H,4)	(H,5)	(H,6)

Venn diagrams

A Venn diagram visualises relationships among subsets of the sample space, especially when more than two events are involved.

Using a Venn diagram

Label the probability of each region. Then apply set operations, De Morgan's laws, union formulae and total probability as appropriate.

Probability tables

A probability table represents two sets of mutually exclusive and exhaustive events.

Event	$A$	$A'$	Total Probability
$B$	$w$	$x$	$w+x$
$B'$	$y$	$z$	$y+z$
Total Probability	$w+y$	$x+z$	$w+x+y+z=1$

Marginal and intersection probabilities

\[P(A)=w+y,\quad P(A')=x+z,\quad P(B)=w+x,\quad P(B')=y+z\]\[P(A\cap B)=w,\qquad P(A'\cap B')=z\]

Conditional probabilities from the table

\[P(B\mid A)=\frac{P(A\cap B)}{P(A)}=\frac{w}{w+y},\qquad P(B'\mid A)=\frac{y}{w+y}\]\[P(B\mid A')=\frac{x}{x+z},\qquad P(B'\mid A')=\frac{z}{x+z}\]

Exam check: for $P(B\mid A)$, divide by the total of the $A$ column, not by $1$. The given event fixes the denominator.

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Event	\(A\)	\(A'\)	Total Probability
\(B\)	\(w\)	\(x\)	\(w+x\)
\(B'\)	\(y\)	\(z\)	\(y+z\)
Total Probability	\(w+y\)	\(x+z\)	\(w+x+y+z=1\)