Probability (simple, compound, simulation) Knowledge-Based Study

Study notes

1. Basic Concepts of Probability

   Probability measures the likelihood that a particular event will occur. The set of all possible outcomes of an experiment is called the sample space, usually denoted by S. An event is any subset of the sample space.

   When all outcomes in a sample space are equally likely, the theoretical probability of an event $E$ is calculated as:

   $$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

   Key Properties

   • The probability of any event is always between 0 and 1, inclusive: $0 \le P(E) \le 1$.
   • An event with a probability of 0 is impossible, while an event with a probability of 1 is certain.
   • The sum of the probabilities of all distinct possible outcomes in a sample space equals 1.
   • The complement of an event $E$, denoted $E'$ or $E^c$, consists of all outcomes not in $E$. Its probability is given by $P(E') = 1 - P(E)$.

   Combining Multiple Events

   A compound event involves the occurrence of two or more simple events. To find compound probabilities, we consider whether events affect one another and whether they can occur simultaneously.

   Independent vs. Dependent Events

   • Independent Events: Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other. For independent events, the multiplication rule is: $$P(A \text{ and } B) = P(A) \times P(B)$$
   • Dependent Events: If the occurrence of $A$ changes the probability of $B$, the events are dependent. We account for this using conditional probability: $$P(A \text{ and } B) = P(A) \times P(B|A)$$ where $P(B|A)$ represents the probability of $B$ occurring given that $A$ has already occurred.

   The Addition Rule

   To determine the probability that event $A$ or event $B$ occurs:

   • If the events are mutually exclusive (cannot happen at the same time), then: $$P(A \text{ or } B) = P(A) + P(B)$$
   • If the events are inclusive (can happen together), we subtract the overlap to avoid double counting: $$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

   Experimental Probability and the Law of Large Numbers

   Experimental probability (or relative frequency) is determined by conducting trials and recording results:

   $$P_{\text{exp}}(E) = \frac{\text{Frequency of event } E}{\text{Total number of trials}}$$

   According to the Law of Large Numbers, as the number of trials increases, the experimental probability gets closer and closer to the theoretical probability.

   Designing a Simulation

   A simulation models a real-world probability situation using tools such as coins, dice, spinners, or random number generators. To conduct a valid simulation:

   1. Identify the component: Determine the basic repeatable event being modeled.
   2. Assign outcomes: Match the theoretical probabilities to random device outcomes (for instance, assigning digits 0–4 to represent a 50% chance in a random digit table).
   3. Establish a trial: Define what constitutes a complete run representing the real-world event.
   4. Perform repeated trials: Execute many trials to record the frequency of success.
   5. Calculate the estimate: Divide total successful trials by the overall number of trials to estimate the likelihood of the event.