Probability Formulae

• Mutually exclusive events are events that cannot happen at the same time; if one occurs, the other cannot.

• The Addition Law (often called the 'OR' rule) states that for mutually exclusive events $A$ and $B$, the probability of either occurring is the sum of their individual probabilities.

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

• If events are not mutually exclusive, this formula must be adjusted by subtracting the overlap ($P(A \text{ and } B)$) to avoid double-counting the shared outcomes.

• This principle is frequently used when calculating the total probability of several distinct successful outcomes in a single trial.

• Independent events are those where the occurrence of one event has no effect on the probability of the other occurring.

• The Multiplication Law (often called the 'AND' rule) is used to find the probability of two independent events both happening.

Key Formula: $P(A \text{ and } B) = P(A) \times P(B)$

• This rule can be extended to any number of independent events, such as $P(A \text{ and } B \text{ and } C) = P(A) \times P(B) \times P(C)$.

• Testing for independence involves checking if the product of individual probabilities equals the probability of the combined event; if they do not match, the events are dependent.

• Conditional probability refers to the likelihood of an event occurring given that another event has already occurred, denoted as $P(B|A)$.

• If events are independent, then $P(B|A) = P(B)$, because the occurrence of $A$ does not change the likelihood of $B$.

• For dependent events, the probability of the second event must be updated based on the outcome of the first, which is often visualized using tree diagrams.

• The general relationship for any two events is $P(A \text{ and } B) = P(A) \times P(B|A)$, which rearranges to the standard conditional formula: $P(B|A) = \frac{P(A \text{ and } B)}{P(A)}$.

• It is critical to distinguish between Mutually Exclusive and Independent events, as they are often confused but represent different logical relationships.

• Note that if two events are mutually exclusive and have non-zero probabilities, they cannot be independent because knowing one happened tells you the other definitely did not.

• Identify the Keyword: Look for 'or' to signal addition and 'and' to signal multiplication, but always verify if the conditions (mutually exclusive or independent) are met first.

• The Sum to One Rule: Always check if a set of events is exhaustive; if so, their probabilities must add up to exactly $1$. This is a powerful tool for finding missing values.

• Independence Testing: If a question asks you to 'show' or 'determine' if events are independent, calculate $P(A) \times P(B)$ and compare it to the given $P(A \text{ and } B)$.

• Sanity Check: Ensure all calculated probabilities are between $0$ and $1$. If you get a value like $1.2$ or $-0.1$, you have likely used the wrong formula or made an arithmetic error.