Basic Probability

• Theoretical probability is calculated when all outcomes in a sample space are equally likely, meaning no outcome is biased or preferred over another.

• The formula for the probability of event $A$, denoted as $P(A)$, is the ratio of the number of successful outcomes to the total number of possible outcomes in the sample space.

Key Formula: $P(A) = \frac{n(A)}{n(S)}$ where $n(A)$ is the number of outcomes in the event and $n(S)$ is the total number of outcomes in the sample space.

• Probabilities can be expressed as fractions, decimals, or percentages (e.g., $1/4$, $0.25$, or $25\%$), but they must never exceed $1$ or $100\%$.

• The complement of an event $A$ (often written as $A'$ or "not $A$") consists of all outcomes in the sample space that are not part of event $A$.

• Because an event must either happen or not happen, the sum of the probability of an event and its complement is always exactly $1$.

• This relationship is used to find missing probabilities: $P(\text{not } A) = 1 - P(A)$.

• In any experiment, the sum of the probabilities of all distinct, mutually exclusive outcomes in the sample space must equal $1$.

• Two events are mutually exclusive if they cannot occur at the same time (e.g., a single card cannot be both a Heart and a Spade).

• For mutually exclusive events $A$ and $B$, the probability of either $A$ or $B$ occurring is the sum of their individual probabilities: $P(A \text{ or } B) = P(A) + P(B)$.

• If events are not mutually exclusive, this simple addition rule does not apply because the overlapping outcomes would be counted twice.

• Relative frequency (or experimental probability) is an estimate of probability based on observed data from an experiment: $\text{Relative Frequency} = \frac{\text{Number of successful trials}}{\text{Total number of trials}}$.

• According to the Law of Large Numbers, as the number of trials increases, the relative frequency tends to get closer to the true theoretical probability.

• Expected frequency is the predicted number of times an event will occur over a specific number of future trials, calculated by multiplying the probability by the number of trials.

Key Formula: $\text{Expected Frequency} = P(\text{event}) \times \text{Total Trials}$

• Check your bounds: Always ensure your final probability is between $0$ and $1$. If you get a negative number or a number greater than $1$, an error has occurred in your calculation.

• Fraction Simplification: Unless specified, probabilities are often best left as fractions. However, always check if the question requires the answer in a specific format like a decimal or percentage.

• Identify Mutually Exclusive Events: Before adding probabilities, verify that the events cannot happen simultaneously to avoid double-counting.

• Sanity Check: If an event is described as 'likely', your calculated probability should be greater than $0.5$. If it is 'unlikely', it should be less than $0.5$.