Probabilities of Single Events

• The Probability Scale: All probabilities are bounded by the interval $[0, 1]$. A probability of $0$ indicates an impossible event, while a probability of $1$ indicates a certain event.

• Sum of Probabilities: The sum of the probabilities of all distinct outcomes in a sample space must exactly equal $1$. This represents the certainty that something within the sample space must happen.

• Notation: The probability of event $A$ occurring is written as $P(A)$. The condition $0 \le P(A) \le 1$ must always hold true for any valid probability model.

Equally Likely Outcomes

• When every outcome in the sample space has the same chance of occurring (e.g., a fair coin or fair die), use the Classical Probability Formula:

$P(A) = \frac{\text{Number of outcomes in event } A}{\text{Total number of outcomes in sample space}}$

Non-Equally Likely Outcomes

• If outcomes are weighted differently (e.g., a biased die or weather patterns), the probability of an event is found by summing the individual probabilities of all outcomes contained within that event.

• $P(A) = \sum P(\text{outcome}_i)$ for all $i$ in event $A$.

• Definition: The complement of event $A$ (denoted $A'$ or $A^C$) consists of all outcomes in the sample space that are not in $A$.

• The Complement Rule: Since an event either happens or it does not, the sum of the probability of an event and its complement is always $1$.

Formula: $P(A') = 1 - P(A)$

• This rule is particularly useful when calculating the probability of 'at least one' or when the complement is easier to count than the event itself.

• Sanity Check: Always ensure your final answer is between 0 and 1. If you calculate a probability of 1.2 or -0.1, a calculation error has occurred.

• Total Probability: Verify that the sum of all probabilities in your distribution equals exactly 1. This is a common diagnostic tool for checking your work.

• Keyword Recognition: Words like 'not', 'except', or 'at least' often signal that using the Complement Rule will be faster than direct calculation.

• Precision: Keep probabilities as exact fractions unless the question specifically requests a decimal. If using decimals, do not round prematurely during intermediate steps.