Probability is a fundamental concept in statistics that helps us understand the likelihood of different events occurring. Within probability theory, there are three key types of probabilities: joint, marginal, and conditional probabilities.

• Marginal Probability refers to the probability of a single event occurring, without considering any other events.
• Joint Probability is the probability of two or more events happening at the same time. It is the probability of the intersection of these events.
• Conditional Probability deals with the probability of an event occurring given that another event has already occurred.

In this article, we will discuss these probabilities in detail, including examples and differences between them as well.

Probability of an Event

Probability of an event quantifies how likely it is for that event to occur. It is a measure that ranges from 0 to 1, where 0 indicates the event cannot happen and 1 indicates the event is certain to happen.

The probability of an event A, denoted as P(A), is defined as:

[Tex]P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}[/Tex]

Sample Space (S)

The set of all possible outcomes of a random experiment. For example, if you roll a die, the sample space S is {1, 2, 3, 4, 5, 6}.

Event (A)

A subset of the sample space is called event in probability.

Event is the specific outcome or set of outcomes that we are interested in. For instance, getting an even number when rolling a die is an event A = {2, 4, 6}.

Joint Probability

Joint probability is the probability of two (or more) events happening simultaneously. It is denoted as P(A∩B) for two events A and B, which reads as the probability of both A and B occurring.

For two events A and B, the joint probability is defined as:

[Tex]P(A \cap B) = P(\text{both } A \text{ and } B \text{ occur})[/Tex]

Note: If A and B are dependent, the joint probability is calculated using conditional probability

Examples of Joint Probability

Rolling Two Dice

• Let A be the event that the first die shows a 3.
• Let B be the event that the second die shows a 5.

The joint probability P(A∩B) is the probability that the first die shows a 3 and the second die shows a 5. Since the outcomes are independent,

P(A∩B) = P(A) ⋅ P(B).

Given: P(A) = 1/6 and P(B) = 1/6, so

⇒ P(A∩B) = 1/6 × 1/6 = 1/36​.

Marginal Probability

Marginal probability refers to the probability of an event occurring, irrespective of the outcomes of other variables. It is obtained by summing or integrating the joint probabilities over all possible values of the other variables.

For two events A and B, the marginal probability of event A is defined as:

[Tex]P(A) = \sum_{B} P(A, B)[/Tex]

Where P(A, B) is the joint probability of both events A and B occurring together. If the variables are continuous, the summation is replaced by integration:

[Tex]P(A) = \int_{B} P(A, B) \, dB[/Tex]

Examples of Marginal Probability

Consider a table showing the joint probability distribution of two discrete random variables X and Y: