Probabilities & Data

Sample Space ($S$): The set of all possible outcomes of a random experiment. For example, in a coin toss, the sample space is $\{H, T\}$.

Event ($E$): A subset of the sample space representing a specific outcome or group of outcomes. The probability of an event is calculated as $P(E) = \frac{n(E)}{n(S)}$, where $n(E)$ is the number of favorable outcomes and $n(S)$ is the total number of outcomes in the sample space.

Probability Scale: All probabilities exist on a closed interval $[0, 1]$. A probability of $0$ indicates an impossible event, while a probability of $1$ indicates a certain event.

Complementary Events: The probability that an event does NOT occur is denoted as $P(E')$. It is calculated using the identity $P(E') = 1 - P(E)$.

Mean ($\bar{x}$): The arithmetic average of a dataset, calculated by summing all values and dividing by the count ($n$). It is highly sensitive to outliers, which can pull the mean toward extreme values.

Median: The middle value of a dataset when ordered from least to greatest. It is a robust measure of center because it remains unaffected by extreme outliers.

Interquartile Range (IQR): A measure of statistical dispersion representing the range of the middle $50\%$ of the data. It is calculated as $IQR = Q_3 - Q_1$, where $Q_3$ is the 75th percentile and $Q_1$ is the 25th percentile.

Standard Deviation ($\sigma$): A measure of how much the data values deviate from the mean. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

Addition Rule (OR): For any two events $A$ and $B$, the probability that at least one occurs is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If the events are mutually exclusive, $P(A \cap B) = 0$.

Multiplication Rule (AND): For independent events, the probability of both occurring is $P(A \cap B) = P(A) \times P(B)$. Independence means the occurrence of one event does not affect the probability of the other.

Conditional Probability: The probability of event $A$ occurring given that event $B$ has already occurred is denoted $P(A|B) = \frac{P(A \cap B)}{P(B)}$. This is essential for analyzing dependent events.

The 'Sum to One' Check: Always verify that the sum of probabilities for all mutually exclusive outcomes in a sample space equals exactly $1$. If it doesn't, an outcome has been missed or a calculation is wrong.

Identifying Outliers: Use the $1.5 \times IQR$ rule. A value is typically considered an outlier if it is less than $Q_1 - 1.5(IQR)$ or greater than $Q_3 + 1.5(IQR)$.

Contextual Reasoning: When asked to choose between mean and median, look at the data distribution. If the data is skewed or contains outliers, the median is usually the better representation of the 'typical' value.

Rounding Precision: In probability, avoid rounding intermediate steps. Keep values as fractions where possible to maintain exactness until the final answer.