Basic Principle of Probability and Counting

Introduction to Probability

• Definition of Probability: The extent to which something is likely to happen; fundamental to probability theory and its applications.

• Usage in Probability Distribution: Understanding the possibility of outcomes for random experiments.

Examples of Probability

• Weather Forecast: "There is a 90% chance of rain" indicates the likelihood of rain.

• Medical Outcome: "There is a 35% chance of successful surgery" represents a prediction of success.

Probability Experiment

• Definition: An action or trial yielding specific results such as counts, measurements, or responses.

• Outcome: Result of a single trial in a probability experiment.

• Sample Space: Set of all possible outcomes from a probability experiment.

• Event: A subset of the sample space. Can include one or more outcomes.

Identifying the Sample Space

• Example: Tossing a coin and rolling a six-sided die.

  • Coin Outcomes: Head (H) or Tail (T) - 2 possible outcomes.

  • Die Outcomes: 1, 2, 3, 4, 5, or 6 - 6 possible outcomes.

• Visual Representation: Tree diagram branches to illustrate outcomes.

• Sample Space Creation: Example shows 12 total outcomes: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.

Events

• Event Definition: A subset of outcomes; may include one or multiple outcomes.

• Simple Event: Consists of a single outcome.

  • Example: Tossing heads and rolling a 3 is represented as A = (H3).

• Complex Event: Consists of multiple outcomes.

  • Example: Tossing heads and rolling an even number is B = (H2, H4, H6).

Identifying Simple Events

• Quality Control Example: Selecting a specific defective machine part (one outcome; simple event).

• Die Roll Example: Event B involves rolling numbers 4, 5, or 6 (three outcomes; not simple).

The Fundamental Counting Principle

• Statement: If one event can occur in m ways and another can occur in n ways, the total outcomes for both is $m imes n$.

• Extension: This principle applies to any number of events occurring in sequence.

Types of Probability

• Types: 3 Main Types:

  1. Classical Probability

  2. Empirical Probability

  3. Subjective Probability

• Probability Notation: The probability of event E is denoted as P(E).

Classical Probability

• Definition: Used when each outcome in a sample space is equally likely to occur.

• Calculation: Formula for classical probability of an event E is not provided in this excerpt.

Empirical Probability

• Definition: Based on observations from probability experiments; described as the relative frequency of an event.

Law of Large Numbers

• Statement: As an experiment is repeated, the empirical probability approaches the theoretical probability.

• Example: Tossing a coin 10 times may yield an empirical probability of 0.3. With several thousand tosses, empirical probability approaches $rac{1}{2}$.

Subjective Probability

• Definition: Based on intuition, educated guesses, and estimates.

• Examples:

  • A physician estimating a 90% chance of full recovery.

  • An analyst predicting a 0.25 chance of an employee strike.

Range of Probabilities Rule

• Rule: The probability of an event E ranges between 0 and 1 (inclusive).

  • Certain Event (P=1): The event is certain to occur.

  • Impossible Event (P=0): The event cannot occur.

  • Even Chance (P=0.5): Represents equal likelihood of occurring.

• Unusual Event: Defined as an event with a probability of 0.05 or less, typically considered highly unlikely.

The Complement of Event E

• Definition: The set of all outcomes in the sample space not included in event E.

• Notation: Denoted by E' or (E) prime.

Conditional Probability and Multiplication Rule

• Definition: Probability of an event occurring, given another event has occurred.

• Notation: P(B | A) is read as the probability of B given A.

Finding Conditional Probabilities

• Example: Two cards drawn from a deck.

  1. Probability the second card is a queen, given the first is a king (not replaced).

  2. The probability of a child having a high IQ given they possess a certain gene.

Solutions 1:

• Remaining deck has 51 cards with 4 queens: $P(B|A) ext{ approximately } = 0.078$.

Solutions 2:

• Sample space of 72 children with 33 high IQs: $P( ext{high IQ}| ext{gene}) ext{ approximately } = 0.458$.

Independent and Dependent Events

• Definition of Independent Events: One event does not affect the other.

• Example: Rolling a die has no effect on coin toss probability.

• Significance: Understanding independence is important in various fields including marketing, medicine, and psychology.

Classification of Events

• To determine independence of two events:

  1. Selecting a king and then a queen (dependent).

  2. Tossing a coin and rolling a die (independent).

  3. Speeding offense impacting car accident likelihood (dependent).

The Multiplication Rule

• Definition: To find the probability of two events occurring sequentially, use the multiplication rule.

• Simplified Rule for Independent Events: If A and B are independent, the probability can be expressed as $P(A) imes P(B)$.

Using the Multiplication Rule to Find Probabilities

• Examples: 1. Select two cards without replacement for probability of king and then queen (dependent event).

• Examples: 2. Toss a coin and roll a die for probability of head then 6 (independent event).

Mutually Exclusive Events

• Definition: Two events are mutually exclusive if they cannot occur simultaneously.

• Event Types: Denoted as P(A and B) for simultaneous occurrences and P(A or B) for one or the other.

Venn Diagram Representation

• Used to illustrate relationships between mutually exclusive events and non-exclusive events.

Deciding Mutually Exclusive Events

• Example 1: Rolling a 3 and rolling a 4: mutually exclusive.

• Example 2: Selecting a male nursing major: not mutually exclusive.

• Example 3: Selecting a female donor and a donor with type O blood: not mutually exclusive.

The Addition Rule for Probability of A or B

• Rule Statement: $P(A ext{ or } B)$ formulated depending on mutual exclusivity. If mutually exclusive, formula simplifies to $P(A) + P(B)$.

Using the Addition Rule to Find Probabilities

• Example 1: Probability of drawing a 4 or an ace from a card deck (mutually exclusive).

• Example 2: Rolling less than 3 or rolling an odd number (not mutually exclusive).

• Solution Example: 2 probabilities computed and shown through Venn diagrams.

References

• Admin. “Probability in Maths - Definition, Formula, Types, Problems and Solutions.” BYJUS, BYJU’S, 4 Oct. 2022, byjus.com/maths/probability/.

• Comment, et al. “Probability in Maths: Formula, Theorems, Definition, Types, Examples.” GeeksforGeeks, 5 Dec. 2024, www.geeksforgeeks.org/probability-in-maths/.