Calculating Probabilities & Events

• The Probability Scale: All probabilities are expressed as values between $0$ and $1$ inclusive. A probability of $0$ indicates an impossible event, while a probability of $1$ indicates an event that is certain to occur.

• Equally Likely Outcomes: If every outcome in a sample space has the same chance of occurring, the probability of an event $A$ is calculated as $P(A) = \frac{n(A)}{n(S)}$, where $n(A)$ is the number of successful outcomes and $n(S)$ is the total number of outcomes.

• The Complement Rule: The probability of an event not occurring, denoted as $P(A')$, is found by subtracting the probability of the event from $1$. This is based on the principle that an event must either happen or not happen, so $P(A) + P(A') = 1$.

• The Multiplication Rule (AND): For independent events, the probability of both $A$ and $B$ occurring is the product of their individual probabilities: $P(A \cap B) = P(A) \times P(B)$. This rule is applied when the outcome of the first event does not influence the likelihood of the second.

• The Addition Rule (OR): For mutually exclusive events, the probability of either $A$ or $B$ occurring is the sum of their individual probabilities: $P(A \cup B) = P(A) + P(B)$. If the events are not mutually exclusive, the intersection must be subtracted to avoid double-counting.

• Tree Diagrams: These are used for sequential events. Probabilities are written on branches, and you multiply along a path to find the probability of a specific sequence of outcomes (AND), then add the results of different paths to find the total probability of a combined event (OR).

Independence vs. Mutual Exclusivity

• Independent Events: These occur when the outcome of one event has no effect on the probability of another. For example, tossing a coin and rolling a die are independent because the coin result doesn't change the die's odds.
• Mutually Exclusive Events: These are events that cannot happen at the same time. If you roll a single die, the events 'rolling a 2' and 'rolling a 5' are mutually exclusive because the die cannot show both numbers simultaneously.

• Keyword Identification: Always look for the words 'AND' and 'OR' in the question. 'AND' usually signals a need for multiplication (intersection), while 'OR' signals a need for addition (union).

• The 'At Least' Shortcut: When a question asks for the probability of 'at least one' success, it is often much faster to calculate $1 - P(\text{no successes})$ than to sum all the individual successful scenarios.

• Sanity Checks: Ensure your final answer is between $0$ and $1$. If you calculate a probability greater than $1$, you likely added probabilities that were not mutually exclusive without subtracting the intersection.

• Tree Diagram Verification: In any tree diagram, the sum of the probabilities on any set of branches originating from a single point must always equal $1$.

• Confusing Independence with Mutual Exclusivity: Students often think that if events are mutually exclusive, they are independent. In fact, if two events are mutually exclusive and $P(A) > 0$, they are highly dependent because the occurrence of one completely prevents the other.

• Forgetting to Subtract the Intersection: When calculating $P(A \cup B)$ for events that are not mutually exclusive, failing to subtract $P(A \cap B)$ leads to an overestimation of the probability.

• Rounding Errors: In multi-step probability problems, especially with tree diagrams, rounding intermediate values can lead to significant errors in the final answer. Keep values as fractions or high-precision decimals until the end.