Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. It provides a quantitative description of the uncertainty associated with various phenomena. Understanding probability is essential for making informed decisions based on data.

An experiment is any process or action that generates a set of outcomes. For example, rolling a die or flipping a coin are common experiments in probability.

The sample space is the set of all possible outcomes of an experiment. For example:

• For a coin toss, the sample space is \( S = { \text{Heads}, \text{Tails} } \).
• For rolling a six-sided die, the sample space is \( S = { 1, 2, 3, 4, 5, 6 } \).

An event is a subset of the sample space. It represents one or more outcomes of an experiment. For example:

• Getting a head when flipping a coin is an event: \( E = { \text{Heads} } \).
• Rolling an even number on a die is an event: \( E = { 2, 4, 6 } \).

The probability of an event is a measure of the likelihood that the event will occur. It is a number between 0 and 1, inclusive, where:

• 0 indicates the event will not occur.
• 1 indicates the event will certainly occur.

The probability of an event \( E \) is calculated as: \[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

The complement of an event \( E \) is the set of outcomes in the sample space that are not in \( E \). It is denoted as \( E' \) or \( E^c \). The probability of the complement of \( E \) is: \[ P(E') = 1 - P(E) \]

Two events are mutually exclusive if they cannot occur at the same time. For example, when rolling a die, the events "rolling a 2" and "rolling a 5" are mutually exclusive.

Two events are independent if the occurrence of one does not affect the occurrence of the other. For example, flipping a coin and rolling a die are independent events.

Consider a fair coin toss. The sample space is \( S = { \text{Heads}, \text{Tails} } \).

• Probability of getting Heads: \( P(\text{Heads}) = \frac{1}{2} \)
• Probability of getting Tails: \( P(\text{Tails}) = \frac{1}{2} \)

Consider rolling a fair six-sided die. The sample space is \( S = { 1, 2, 3, 4, 5, 6 } \).

• Probability of rolling a 3: \( P(3) = \frac{1}{6} \)
• Probability of rolling an even number: \( P(\text{Even}) = \frac{3}{6} = \frac{1}{2} \)

A bag contains 3 red balls, 2 blue balls, and 5 green balls. One ball is drawn at random. Calculate the probability of drawing:

1. A red ball
2. A blue ball
3. A green ball

Solution:

1. Total number of balls = 3 + 2 + 5 = 10
2. Probability of drawing a red ball: \( P(\text{Red}) = \frac{3}{10} \)
3. Probability of drawing a blue ball: \( P(\text{Blue}) = \frac{2}{10} = \frac{1}{5} \)
4. Probability of drawing a green ball: \( P(\text{Green}) = \frac{5}{10} = \frac{1}{2} \)

If the probability of raining tomorrow is 0.3, what is the probability that it will not rain?

Solution: \[ P(\text{Not Rain}) = 1 - P(\text{Rain}) = 1 - 0.3 = 0.7 \]

A card is drawn from a standard deck of 52 cards. Calculate the probability of drawing either a King or a Queen.

Solution:

• Number of Kings in the deck = 4
• Number of Queens in the deck = 4
• Total number of favorable outcomes = 4 + 4 = 8
• Probability of drawing a King or a Queen: \( P(\text{King or Queen}) = \frac{8}{52} = \frac{2}{13} \)

In this section, we covered the basic concepts of probability, including experiments, sample spaces, events, and the calculation of probabilities. Understanding these foundational concepts is crucial for more advanced topics in probability and statistics. In the next section, we will explore the rules of probability, which will help us handle more complex probability scenarios.