Probability rules

• The sample space $S$ is the set of all possible outcomes.
• An event is a set of outcomes.
• If all outcomes are equally likely: $P(E)=\frac{n(E)}{n(S)}$

Example. A die is rolled. $S=\{1,2,3,4,5,6\}$. Let $A$ = prime numbers $=\{2,3,5\}$ and $B$ = even numbers $=\{2,4,6\}$.

$P(A)=\frac{3}{6}P(A\text{ and }B)=P(\{2\})=\frac{1}{6}$

For any two events:

$P(A\text{ or }B)=P(A)+P(B)-P(A\text{ and }B)$

Example. With the die above:

$P(\text{prime or even})=\frac{3}{6}+\frac{3}{6}-\frac{1}{6}=\frac{5}{6}$

Example. A card is drawn from a standard deck of 52 cards:

$P(\text{king or heart})=\frac{4}{52}+\frac{13}{52}-\frac{1}{52}=\frac{16}{52}=\frac{4}{13}$

The king of hearts is counted in both, so it is subtracted once.

Events are mutually exclusive if they cannot happen together: $P(A\text{ and }B)=0$. Then

$P(A\text{ or }B)=P(A)+P(B)$

Example. Rolling a 1 and rolling an even number are mutually exclusive.

Test. If $P(A\text{ or }B)=P(A)+P(B)$, the events are mutually exclusive. If $P(A)=0,5$, $P(B)=0,3$ and $P(A\text{ or }B)=0,65$, then $P(A\text{ and }B)=0,15\ne 0$, so they are not.

The complement of $A$ ("not $A$") contains every outcome not in $A$:

$P(\text{not }A)=1-P(A)$

Example. $P(\text{neither a king nor a heart})=1-\frac{4}{13}=\frac{9}{13}$

Complements are the quickest route for "at least one" and "neither" questions.