1  Random Experiment

A random experiment is a process or procedure that leads to one of several possible outcomes. These outcomes are uncertain and can’t be precisely predicted beforehand. Random experiments are fundamental in probability theory and statistics. Here are a few key points about random experiments:

• Uncertainty: The outcome of a random experiment is uncertain. Even if the experiment is conducted under identical conditions, the outcome may differ each time due to inherent randomness.

• Sample Space: A random experiment has a sample space (\(S\) or \(\Omega\)), which is the set of all possible outcomes of the experiment. Each outcome in the sample space is a distinct possible result of the experiment.

• Events: An event is a subset of the sample space. It represents a specific outcome or a combination of outcomes of the random experiment. Events can be simple (involving a single outcome) or compound (involving multiple outcomes).

• Probability: Probability theory is used to assign numerical values to events, representing the likelihood of their occurrence. Probabilities range from 0 (impossible event) to 1 (certain event).

Examples of random experiments include rolling a die, tossing a coin, drawing a card from a deck, conducting a survey, or measuring a random variable. Each of these experiments has a set of possible outcomes, forming the sample space, and the specific outcome that occurs is not known in advance, making it a random experiment.

1.1 Uncertainty

In the context of probability and statistics, Uncertainty refers to the lack of perfect knowledge about a situation or an event. It reflects the unpredictability or ambiguity associated with the outcomes of a random experiment or a process. Uncertainty arises due to various factors, including incomplete information, variability, and randomness.

1.2 Sample Space

In probability theory, the sample space, denoted by \(S\) or \(\Omega\), is the set of all possible outcomes of a random experiment or process.

It encompasses every conceivable result that could occur, and each outcome in the sample space is considered equally likely.

For example, when rolling a fair six-sided die, the sample space consists of the numbers \(1, 2, 3, 4, 5\), and \(6\), as these are all the possible outcomes.

In more complex situations, the sample space can be finite, countably infinite, or uncountably infinite. It is a fundamental concept in probability theory because all probabilities of events are calculated based on the outcomes within the sample space.

1.2.1 Examples of Sample Spaces

1. Rolling a Fair Six-Sided Die:
   • Sample Space: \(\{1, 2, 3, 4, 5, 6\}\)
   • In this experiment, rolling a fair six-sided die can result in any of the numbers 1 through 6. Therefore, the sample space consists of these six possible outcomes.
2. Flipping a Biased Coin:
   • Sample Space: \(\{\text{Heads}, \text{Tails}\}\)
   • If the coin is biased, it might have different probabilities of landing heads or tails. The sample space in this case represents the two possible outcomes: “Heads” or “Tails.”
3. Drawing a Card from a Standard Deck:
   • Sample Space: \(\{\text{Ace of Hearts}, \text{2 of Hearts}, \ldots, \text{King of Spades}\}\) (total 52 cards)
   • When drawing a single card from a standard deck of 52 playing cards, each card represents a unique outcome. The sample space includes all 52 cards in the deck, such as the “Ace of Hearts,” “2 of Hearts,” and so on up to the “King of Spades.”

In each case, the sample space comprises all possible outcomes of the respective random experiment.

1.3 Events

In probability theory, an event is a specific outcome or a set of outcomes of a random experiment or process. Events are subsets of the sample space (\(S\)) and can be any collection of possible outcomes from the sample space. Events are used to describe the occurrence or non-occurrence of particular results in a given experiment.

For example, consider rolling a fair six-sided die. The sample space for this experiment is \(\{1, 2, 3, 4, 5, 6\}\). Here are a few examples of events:

1. Event A: Rolling an even number.
   • This event corresponds to the set of outcomes \(\{2, 4, 6\}\).
2. Event B: Rolling a number greater than 4.
   • This event corresponds to the set of outcomes \(\{5, 6\}\).
3. Event C: Rolling a number less than or equal to 3.
   • This event corresponds to the set of outcomes \(\{1, 2, 3\}\).

Events can be simple (like rolling a 3) or complex (like rolling an even number and getting a result greater than 2). The probability of an event \(P(A)\) is a measure of the likelihood of that event occurring and is calculated based on the number of favorable outcomes for the event divided by the total number of outcomes in the sample space.

2 Probability

Probability is a measure of the likelihood or chance of an event occurring. It quantifies the uncertainty associated with random phenomena and provides a numerical representation of the likelihood of different outcomes.

In formal terms, the probability of an event \(A\), denoted as \(P(A)\), is a number between 0 and 1, where:

• \(P(A) = 0\) indicates that the event \(A\) is impossible.
• \(P(A) = 1\) indicates that the event \(A\) is certain to occur.
• \(0 < P(A) < 1\) indicates that the event \(A\) has some probability between impossible and certain.

The sum of the probabilities of all possible outcomes in a sample space (the set of all possible outcomes of a random experiment) is always equal to 1. Mathematically, if \(S\) represents the sample space, then for any event \(A\) in \(S\):

\[P(A) + P(\text{not } A) = 1\]

Probability theory provides a framework for analyzing random events and making predictions based on uncertain information. It is widely used in various fields, including statistics, mathematics, science, engineering, economics, and social sciences, to model and analyze uncertain situations and make informed decisions.