Calculating Probabilities & Events

Equally Likely Outcomes: When every outcome in a sample space has the same chance of occurring, the probability of an event is calculated by dividing the number of successful outcomes by the total number of possible outcomes. This is the foundation of theoretical probability, often used in games of chance like dice or cards.

The Complement Rule: The complement of an event $A$, denoted as $A'$ or $\bar{A}$, represents the event that $A$ does not happen. Because an event must either happen or not happen, the sum of their probabilities is always $1$, leading to the formula $P(A') = 1 - P(A)$.

Theoretical vs. Experimental: Theoretical probability is based on logic and mathematical models, while experimental probability (relative frequency) is based on the results of actual trials. As the number of trials in an experiment increases, the experimental probability typically converges toward the theoretical probability.

Independent Events: Two events are independent if the outcome of the first has no influence on the probability of the second. To find the probability of both independent events $A$ and $B$ occurring (the 'AND' rule), you multiply their individual probabilities: $P(A \text{ and } B) = P(A) \times P(B)$.

Mutually Exclusive Events: Events are mutually exclusive if they cannot occur at the same time. For such events, the probability of either $A$ or $B$ occurring (the 'OR' rule) is the sum of their individual probabilities: $P(A \text{ or } B) = P(A) + P(B)$.

Tree Diagrams: These are visual tools used to map out sequences of events, especially when probabilities change after the first event (dependent events). You multiply probabilities along the branches to find the probability of a specific path and add the results of different paths to find the total probability of a combined outcome.

Histograms and Frequency Density: In continuous data, probabilities are often derived from histograms where the area of the bar represents the frequency, not the height. The height of the bar is the frequency density, calculated as $\text{Frequency} \div \text{Class Width}$.

Linear Interpolation: When an event falls within a specific class interval of a histogram, we assume the data is spread evenly across that interval. We use the proportion of the class width covered by the event to estimate the corresponding frequency and, subsequently, the probability.

Area as Probability: To find the probability of a value falling within a certain range on a histogram, you calculate the area of the bars (or parts of bars) covering that range and divide it by the total area of the entire histogram.

Keyword Identification: Train yourself to spot the words 'AND' and 'OR' in word problems. 'AND' usually signals a multiplication of probabilities (intersection), while 'OR' signals an addition (union), provided the events are mutually exclusive.

Sanity Checks: Always verify that your final probability is between $0$ and $1$. If you calculate a probability greater than $1$, you likely added probabilities that were not mutually exclusive or forgot to divide by the total outcomes.

The Power of the Complement: If a question asks for the probability of 'at least one' or 'not all', it is often much faster to calculate the probability of the opposite event and subtract it from $1$.

Histogram Accuracy: When working with histograms, always check the axes. If the vertical axis is 'Frequency Density', you must calculate the area of the bars to find the frequencies needed for your probability fraction.