Rules of Probability for Events

When working with probability, there are some basic rules for events that are used again and again.

If the probability of an event is $P(A)$, then the probability that it does not occur is:

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

Example:

The probability of rolling a six with a die is $P(A) = \tfrac{1}{6}$.

The probability of not rolling a six is $1 - \tfrac{1}{6} = \tfrac{5}{6}$.

If two events cannot happen at the same time (they are mutually exclusive), then:

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

Example:

In a coin toss, $P(\text{heads}) = \tfrac{1}{2}$ and $P(\text{tails}) = \tfrac{1}{2}$.

The probability of getting either heads or tails is $\tfrac{1}{2} + \tfrac{1}{2} = 1$.

If two events can happen at the same time, you must subtract the overlap once:

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

Example:

In a card game, the probability of drawing a heart is $P(A)$, and the probability of drawing a face card is $P(B)$.

A heart face card counts in both groups, so it must be subtracted once.

If two events are independent, then:

$$ P(A \text{ and } B) = P(A) \cdot P(B) $$

Example:

The probability of rolling a six with one die is $\tfrac{1}{6}$.

If you roll twice, the probability of getting two sixes in a row is:

$$ \large \tfrac{1}{6} \cdot \tfrac{1}{6} = \tfrac{1}{36} $$

Sometimes the probability of one event depends on another event already occurring. This is called conditional probability:

$$ P(A|B) = \frac{P(A \text{ and } B)}{P(B)} $$

Example:

If you draw two cards one after the other from a deck without replacement:

The probability that the first is an ace is $\tfrac{4}{52}$.

If that happens, only 51 cards remain, so the probability of another ace is $\tfrac{3}{51}$.

Here the probability of the second draw depends on what happened in the first.

Rule	Formula	Condition
Complement	$$ P(\text{not A}) = 1 - P(A) $$	-
Addition	$$ P(A \text{ or } B) = P(A) + P(B) $$	If A and B cannot happen simultaneously
Generalized addition	$$ P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B) $$	If A and B can overlap
Multiplication	$$ P(A \text{ and } B) = P(A) \cdot P(B) $$	If A and B are independent
Conditional probability	$$ P(A\|B) = \tfrac{P(A \text{ and } B)}{P(B)} $$	-