Basic Concepts of Probability and Counting

Probability Experiments

A probability experiment is an action or trial through which specific results (counts, measurements, or responses) are obtained. The result of a single trial is called an outcome. The sample space is the set of all possible outcomes of a probability experiment. An event consists of one or more outcomes and is a subset of the sample space.

• Simple event: An event that consists of a single outcome.

• Compound event: An event that consists of more than one outcome.

Example: If you roll a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Rolling a 3 is a simple event; rolling an even number (2, 4, 6) is a compound event.

The Fundamental Counting Principle

Definition and Application

The Fundamental Counting Principle states that if one event can occur in m ways and a second event can occur in n ways, then the number of ways the two events can occur in sequence is m × n. This principle can be extended for any number of events occurring in sequence.

• For k events, with n_1, n_2, ..., n_k possible outcomes respectively, the total number of outcomes is .

Example: If you are purchasing a car and there are 3 manufacturers, 2 car sizes, and 4 colors, the total number of possible selections is .

Example: For a car's security system with a 4-digit code (digits 0-9):

• If digits can be repeated: possible codes.

• If digits cannot be repeated: possible codes.

• If the first digit cannot be 0 or 1 (but digits can repeat): possible codes.

Types of Probability

Classical (Theoretical) Probability

Classical probability is used when all outcomes in the sample space are equally likely. The probability of event E is:

Example: Rolling a six-sided die:

• Probability of rolling a 3:

• Probability of rolling a 7: (since 7 is not in the sample space)

• Probability of rolling less than 5:

Empirical (Statistical) Probability

Empirical probability is based on observations from experiments. It is calculated as:

Example: In a survey of 1,502 adults, 560 read only print books. The probability that the next adult surveyed read only print books is .

Subjective Probability

Subjective probability is based on intuition, educated guesses, or estimates. For example, a doctor may estimate a 90% chance of recovery for a patient based on experience.

Classifying Probability Statements

• Classical: Probability of winning a 1,000-ticket raffle with one ticket is (all outcomes equally likely).

• Empirical: Probability that a randomly chosen voter is under 35 is 0.3 (based on survey data).

• Subjective: Probability of getting an A on the next test is 0.9 (based on personal judgment).

• 

Complementary Events

Definition and Calculation

The complement of event E, denoted E', is the set of all outcomes in the sample space that are not in E. The probabilities of an event and its complement always sum to 1:

Thus,

Example: If the probability that a user is 25 to 34 years old is 0.255, then the probability that a user is not 25 to 34 years old is .

Range of Probabilities Rule

The probability of any event E is always between 0 and 1, inclusive:

• P(E) = 0: Impossible event

• P(E) = 1: Certain event

Tree Diagrams and Probability

Using Tree Diagrams

Tree diagrams are visual tools used to display all possible outcomes of a probability experiment, especially when events occur in sequence. Each branch represents a possible outcome at each stage.

Example: Tossing a coin and spinning a spinner numbered 1 to 8. The tree diagram shows all possible pairs (e.g., H1, H2, ..., T8).

• Event A: Tossing a tail and spinning an odd number = {T1, T3, T5, T7}

• Event B: Tossing a head or spinning a number greater than 3 = {H1, H2, H3, H4, H5, H6, H7, H8, T4, T5, T6, T7, T8}

Law of Large Numbers

The Law of Large Numbers states that as an experiment is repeated many times, the empirical probability of an event approaches its theoretical probability.

Summary Table: Types of Probability