sample space

A sample space is a collection of all possible outcomes of a random experiment. It is the set of all possible results of a random process, or a set of possible values of a random variable. The sample space provides a framework for understanding probability and statistical analysis, as it represents all the possible outcomes of an event. The elements of a sample space are known as sample points or outcomes. For example, if you roll a dice, the sample space would be the set {1, 2, 3, 4, 5, 6}.

Events and Probability

In probability theory, an event is a set of outcomes of a random experiment. It is a collection of one or more possible outcomes from a sample space. An event can be either simple, consisting of a single outcome, or it can be complex, consisting of multiple outcomes. The probability of an event is a measure of the likelihood that the event will occur, expressed as a number between 0 and 1, where 0 represents that the event is impossible and 1 represents that the event is certain to occur.

The probability of an event can be calculated using the following formula:

P(A) = Number of favorable outcomes / Total number of possible outcomes

where A is the event of interest, and the numerator and denominator are taken from the sample space.

There are two types of events: mutually exclusive and non-mutually exclusive events. Mutually exclusive events are events that cannot occur at the same time, and their sample spaces do not overlap. For example, the event “rolling a 4 on a die” and the event “rolling a 5 on a die” are mutually exclusive events because they cannot occur simultaneously. On the other hand, non-mutually exclusive events are events that can occur simultaneously. For example, the event “rolling an even number on a die” and the event “rolling a number greater than 4 on a die” are non-mutually exclusive events because they can occur at the same time.

In addition, the concept of conditional probability is used to describe the probability of an event given that another event has occurred. The formula for conditional probability is:

P(A|B) = P(A and B) / P(B)

where A and B are two events and P(A and B) is the probability of both events occurring simultaneously.

Probability theory is an important tool in many fields, including statistics, economics, finance, and engineering, and it plays a crucial role in understanding and modeling random processes

Experiments and random experiments

An experiment is an activity that is performed with the intention of observing the result. A random experiment is a type of experiment in which the outcome is not determined by any predetermined pattern or rule, but instead is influenced by chance. The outcome of a random experiment is uncertain, and can be any one of a set of possible outcomes.

Examples of random experiments include flipping a coin, rolling a dice, drawing a card from a deck, and measuring the height of a randomly selected person. In each of these examples, there is a set of possible outcomes and the actual outcome is not determined in advance, but instead is determined by chance.

The results of a random experiment are modeled using a sample space, which is the set of all possible outcomes of the experiment. Each outcome in the sample space is assigned a probability, which represents the likelihood that the outcome will occur. The sum of the probabilities of all outcomes in the sample space is equal to 1, which means that the sum of the probabilities of all possible outcomes of a random experiment is 1.

The study of random experiments and their outcomes is known as probability theory, which is an important branch of mathematics that is widely used in many fields, including statistics, finance, and engineering. Probability theory provides a framework for understanding and modeling random processes and for making predictions about the outcomes of random experiments

Ideas of deterministic and non-deterministic experiments

A deterministic experiment is an experiment in which the outcome is completely determined by the initial conditions and the underlying rules governing the experiment. In a deterministic experiment, the same initial conditions and rules will always produce the same outcome. For example, calculating the product of 2 multiplied by 3 is a deterministic experiment because the outcome (6) is determined by the initial conditions (the numbers 2 and 3) and the rule for multiplication.

In contrast, a non-deterministic experiment is an experiment in which the outcome is not determined by the initial conditions and rules. The outcome of a non-deterministic experiment is influenced by chance or randomness, and the same initial conditions and rules can produce different outcomes. For example, flipping a coin is a non-deterministic experiment because the outcome (heads or tails) is not determined by the initial conditions (the coin) and the rule for flipping, but instead is influenced by chance.

 Definition of sample space

A sample space is the set of all possible outcomes of a random experiment. The sample space provides a framework for understanding and modeling probability and random processes, as it represents all the possible outcomes of an event. The sample space is used to assign probabilities to the individual outcomes, which represent the likelihood that each outcome will occur.

discrete sample space

A discrete sample space is a sample space in which the possible outcomes are finite and can be listed or enumerated. The sample space is called “discrete” because the outcomes are separated or distinct from each other. For example, the sample space for rolling a six-sided die is {1, 2, 3, 4, 5, 6}, which is a discrete sample space because the possible outcomes are finite and can be listed.

In contrast, a continuous sample space is a sample space in which the possible outcomes are not finite and cannot be listed. Instead, the outcomes form a continuous range of values. For example, the sample space for measuring the height of a randomly selected person is a continuous sample space because the possible heights form a continuous range of values.

events

An event is a set of outcomes of a random experiment. It represents a collection of one or more possible outcomes from the sample space. In probability theory, events are used to model and analyze the outcomes of random experiments.

There are several types of events, including:

1. Simple events: A simple event is an event that consists of a single outcome from the sample space. For example, rolling a 6 on a six-sided die is a simple event.
2. Compound events: A compound event is an event that consists of multiple outcomes from the sample space. For example, rolling an even number on a six-sided die is a compound event.
3. Union of events: The union of two or more events is the event that consists of all outcomes that belong to at least one of the events. The symbol for the union of events A and B is A U B. For example, if A represents the event “rolling a 4 on a die” and B represents the event “rolling a 5 on a die”, then A U B represents the event “rolling a 4 or 5 on a die”.
4. Intersection of events: The intersection of two or more events is the event that consists of all outcomes that belong to all of the events. The symbol for the intersection of events A and B is A ∩ B. For example, if A represents the event “rolling an even number on a die” and B represents the event “rolling a number greater than 4 on a die”, then A ∩ B represents the event “rolling a 6 on a die”.
5. Mutually exclusive events: Mutually exclusive events are events that cannot occur at the same time. In other words, they have no common outcomes in their sample spaces. For example, the events “rolling a 4 on a die” and “rolling a 5 on a die” are mutually exclusive because they cannot occur at the same time.
6. Complementary event: The complementary event of an event A is the event that consists of all outcomes from the sample space that do not belong to event A. The symbol for the complement of event A is Ā. For example, if A represents the event “rolling an even number on a die”, then Ā represents the event “rolling an odd number on a die”.
7. Exhaustive events: Exhaustive events are events that together make up the entire sample space. In other words, they are events that cover all possible outcomes of a random experiment. For example, the events “rolling an even number on a die” and “rolling an odd number on a die” are exhaustive because they together make up the entire sample space of rolling a die

Classical definition of probability

The classical definition of probability is a method of defining the probability of an event based on the ratio of the number of favorable outcomes to the number of possible outcomes in the sample space. The classical definition of probability is expressed as follows:

Probability of an event A = Number of favorable outcomes of A / Total number of possible outcomes in the sample space

In other words, the classical definition of probability states that the probability of an event is proportional to the number of favorable outcomes of that event divided by the total number of possible outcomes in the sample space. The classical definition of probability assumes that all outcomes in the sample space are equally likely, so the probability of an event can be calculated by counting the number of favorable outcomes and dividing by the total number of possible outcomes.

For example, if you have a fair six-sided die, the probability of rolling a 4 is 1/6 because there is only 1 favorable outcome (rolling a 4) out of 6 possible outcomes (rolling a 1, 2, 3, 4, 5, or 6)

Definition of conditional probability Definition of independence of two events

The addition theorem of probability states that the probability of the union of two or more events is equal to the sum of the probabilities of the individual events minus the sum of the probabilities of the intersections of the events. The theorem can be expressed as follows:

P(A U B) = P(A) + P(B) – P(A ∩ B)

where A and B are two events and A U B is their union.

For three events A, B, and C, the addition theorem of probability can be expressed as follows:

P(A U B U C) = P(A) + P(B) + P(C) – P(A ∩ B) – P(A ∩ C) – P(B ∩ C) + P(A ∩ B ∩ C)

This theorem provides a useful way to calculate the probability of a compound event in terms of the probabilities of individual events and their intersections

Conditional probability is the probability of an event occurring given that another event has already occurred. It is used to quantify the relationship between two events. The definition of conditional probability is expressed as follows:

P(A | B) = P(A ∩ B) / P(B)

where A and B are two events and P(A | B) is the probability of event A occurring given that event B has already occurred. The symbol “|” is read as “given” or “conditional on”.

The conditional probability of event A given event B is only defined when the probability of event B is greater than zero. This is because event B must have already occurred for the conditional probability to be meaningful.

Independence of two events means that the occurrence of one event does not affect the probability of the other event occurring. In other words, the events are not dependent on each other. Two events A and B are independent if and only if:

P(A | B) = P(A)

In other words, the probability of event A occurring given that event B has already occurred is equal to the probability of event A occurring regardless of the occurrence of event B.

A simple numerical problem involving conditional probability and independence can be stated as follows:

Problem: You have a box containing 3 red balls and 7 blue balls. You choose a ball randomly from the box and then replace it. Then you choose a second ball randomly from the box. What is the probability that the first ball is red and the second ball is blue?

Solution: Let event A be the event that the first ball is red and event B be the event that the second ball is blue. Since the balls are replaced after each draw, the events are independent, so we have:

P(A ∩ B) = P(A) * P(B | A) = (3/10) * (7/10) = 21/100

So the probability that the first ball is red and the second ball is blue is 21/100