Chapter 13: Simple Probability

13.1 Experimental Probability

Experimental probability is the ratio of the number of times an event occurs to the total number of trials in an experiment.

$$\text{Experimental Probability} = \frac{\text{Frequency of event occurrence}}{\text{Total number of trials}}$$

As the number of trials increases, the experimental probability approaches the theoretical probability.

13.2 Probability Theory Involving Equally Likely Outcomes

1. Sample Space and Events

• Sample Space ($S$): The set of all possible outcomes of an experiment. The total number of outcomes in the sample space is denoted as $n(S)$.
• Event ($A$): A subset of the sample space $S$. The number of outcomes in event $A$ is denoted as $n(A)$.

2. Probability Formula

The theoretical probability of an event $A$ occurring with equally likely outcomes is given by:

$$P(A) = \frac{n(A)}{n(S)}$$

Where:

• $n(A)$ = Number of outcomes favorable to event $A$
• $n(S)$ = Total number of possible outcomes in sample space $S$

3. Range of Probability Values

The probability of any event $A$ satisfies:

$$0 \le P(A) \le 1$$

• $P(A) = 0$: An impossible event (will never happen).
• $P(A) = 1$: A certain event (will definitely happen).

13.3 Complement of an Event

The complement of event $A$, denoted as $A'$, is the set of all outcomes in the sample space $S$ that are not in event $A$.

Formula for Complementary Event

$$P(A') = 1 - P(A) \quad \text{or} \quad P(A) + P(A') = 1$$

13.4 Simple Probability Applications

Probability can be expressed as a fraction, decimal, or percentage and is widely used in risk assessment, decision-making, and statistical projections.