Cheatsheet Content
1. Basic Terminology Experiment: An operation that can produce some well-defined outcomes. Random Experiment: An experiment whose outcome cannot be predicted with certainty, but all possible outcomes are known. Sample Space ($S$): The set of all possible outcomes of a random experiment. Event ($E$): A subset of the sample space. Elementary Event: An event having only one sample point. Compound Event: An event having more than one sample point. Impossible Event: An event that cannot occur ($\emptyset$). Sure Event: An event that is certain to occur ($S$). 2. Probability of an Event Classical (A Priori) Definition: $P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of possible outcomes}}$ For any event $E$, $0 \le P(E) \le 1$. $P(\emptyset) = 0$ (Probability of an impossible event). $P(S) = 1$ (Probability of a sure event). 3. Complementary Events Complement of an Event ($E'$ or $\bar{E}$): The event that $E$ does not occur. $P(E') = 1 - P(E)$ $P(E) + P(E') = 1$ 4. Mutually Exclusive Events Two events $A$ and $B$ are mutually exclusive if they cannot occur simultaneously, i.e., $A \cap B = \emptyset$. For mutually exclusive events: $P(A \cap B) = 0$. 5. Exhaustive Events A set of events $E_1, E_2, ..., E_n$ is exhaustive if their union is the sample space, i.e., $E_1 \cup E_2 \cup ... \cup E_n = S$. 6. Mutually Exclusive and Exhaustive Events If $E_1, E_2, ..., E_n$ are mutually exclusive and exhaustive events, then $P(E_1) + P(E_2) + ... + P(E_n) = 1$. 7. Addition Rule of Probability For any two events $A$ and $B$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ For three events $A, B, C$: $P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(B \cap C) - P(C \cap A) + P(A \cap B \cap C)$ For mutually exclusive events $A$ and $B$: $P(A \cup B) = P(A) + P(B)$ 8. Independent Events Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the occurrence of the other. Multiplication Rule for Independent Events: $P(A \cap B) = P(A) \cdot P(B)$ If $A$ and $B$ are independent, then $A$ and $B'$, $A'$ and $B$, and $A'$ and $B'$ are also independent. 9. Odds Odds in favor of an event $E$: $\frac{P(E)}{P(E')} = \frac{P(E)}{1 - P(E)}$ Odds against an event $E$: $\frac{P(E')}{P(E)} = \frac{1 - P(E)}{P(E)}$ 10. Important Concepts Union ($A \cup B$): Event $A$ or $B$ or both occur. Intersection ($A \cap B$): Event $A$ and $B$ both occur. Difference ($A - B$ or $A \cap B'$): Event $A$ occurs but $B$ does not occur. 11. Solved Problem Example Problem: A die is thrown. Find the probability of getting an even number. Solution: Sample Space $S = \{1, 2, 3, 4, 5, 6\}$, so $n(S) = 6$. Let $E$ be the event of getting an even number. $E = \{2, 4, 6\}$, so $n(E) = 3$. $P(E) = \frac{n(E)}{n(S)} = \frac{3}{6} = \frac{1}{2}$.
