Basic Probability and Set Operations

Basic Probability

• Probability (P) is calculated by dividing the number of favorable outcomes by the total number of outcomes.
• Example: In a deck of 52 cards, the probability of drawing a heart is 13/52, since there are 13 hearts and 52 total cards.

Basic Set Operations

• The union of two sets (AUB) represents the probability that an element belongs to either set A or set B or both.
• The intersection of two sets (A∩B) represents the probability that an element belongs to both set A and set B.
• The complement of a set represents the probability that an element is not in a given set.
• Example: If 60 students are enrolled in Mathematics (Set A) and 50 in Physics (Set B) out of 100 students, with 30 in both, the probability of a student being enrolled in either Mathematics or Physics is P(AUB) = P(A) + P(B) - P(A∩B) = 0.6 + 0.5 - 0.3 = 0.8.

Mutually Exclusive Events

• Two events are mutually exclusive if they cannot occur at the same time, meaning their intersection is empty.
• The probability of the union of two mutually exclusive events is the sum of their individual probabilities.
• Example: if events A and B are mutually exclusive, with P(A) = 0.3 and P(B) = 0.4, then the probability of A or B occurring P(AUB) = P(A) + P(B) = 0.3 + 0.4 = 0.7.

Independence

• Two events are independent if the occurrence of one event does not affect the probability of the other event.
• The probability of the intersection of two independent events is the product of their individual probabilities: P(A∩B) = P(A) * P(B).

Total Probability Rule

• The total probability rule calculates the probability of an event by considering different mutually exclusive conditions.
• It involves conditional probabilities: the probability of an event occurring given that another event has already occurred.
• Example: The probability of event D occurring can be calculated as the sum of the conditional probabilities of D given each condition (A and B) multiplied by the respective probabilities of those conditions: P(D) = P(D|A) * P(A) + P(D|B) * P(B).

Conditional Probability

• Conditional probability (P(A|B)) represents the probability of event A occurring given that event B has already occurred.
• It is calculated by dividing the probability of both events occurring (P(A∩B)) by the probability of the event that is already known to occur (P(B)).
• Example: If the probability of both events A and B occurring is P(A∩B) = 0.2 and the probability of event B occurring is P(B) = 0.5, then the conditional probability of A given B is P(A|B) = P(A∩B) / P(B) = 0.2 / 0.5 = 0.4.

Counting Questions (Combinatorics)

• Combinatorics deals with counting the number of ways certain events can occur or arrangements can be made.
• Four main types of problems are:
  • Permutations: order matters.
  • Combinations: order does not matter.
  • Factorials: multiplying integers up to a given number.
  • Multiplication Principle: if one event occurs in m ways and another event occurs in n ways, then both events can occur in m * n ways.

Random Variables

• A random variable is a variable whose value is a numerical outcome of a random phenomenon.
• A cumulative distribution function (CDF) gives the probability that a random variable will take a value less than or equal to a certain value.
• A discrete random variable has a finite number of possible outcomes or a countable number of outcomes.
• The expected value (E(X)) of a discrete random variable X is the sum of the products of each outcome and its probability.