Probability: Joint vs. Marginal vs. Conditional
Last Updated : 11 Mar, 2025
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:
P(A) = \dfrac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
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:
P(A \cap B) = P(\text{both } A \text{ and } B \text{ occur})
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:
P(A) = \sum_{B} P(A, B)
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:
P(A) = \int_{B} P(A, B) \, dB
Examples of Marginal Probability
Consider a table showing the joint probability distribution of two discrete random variables X and Y:
X/Y | Y = 1 | Y = 2 |
|---|
X = 1 | 0.1 | 0.2 |
|---|
X = 2 | 0.3 | 0.4 |
|---|
To find the marginal probability of X = 1:
P(X = 1) = P(X = 1, Y = 1) + P(X = 1, Y = 2) = 0.1 + 0.2 = 0.3
Read More about Marginal Distribution.
Conditional Probability
Conditional probability is the probability of an event occurring given that another event has already occurred. It provides a way to update our predictions or beliefs about the occurrence of an event based on new information.
The conditional probability of event A given event B is denoted as P(A∣B) and is defined by the formula:
P(A|B) = \frac{P(A \cap B)}{P(B)}
Where:
- P(A∩B) is the joint probability of both events A and B occurring.
- P(B) is the probability of event B occurring.
Examples of Conditional Probability
Suppose we have a deck of 52 cards, and we want to find the probability of drawing an Ace given that we have drawn a red card.
- Let A be the event of drawing an Ace.
- Let B be the event of drawing a red card.
There are 2 red Aces in a deck (Ace of hearts and Ace of diamonds) and 26 red cards in total.
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{2}{52}}{\frac{26}{52}} = \frac{2}{26} = \frac{1}{13}
Difference between Joint, Marginal, and Conditional Probability
The key differences between joint, marginal and conditional probability are listed in the following table:
- For Independent Events
- For Dependent Events
| Aspect | Joint Probability | Marginal Probability | Conditional Probability |
|---|
| Definition | The probability of two or more events occurring together. | The probability of a single event irrespective of the occurrence of other events. | The probability of an event given that another event has occurred. |
| Notation | P(A∩B) or P(A, B) | P(A) or P(B) | P(A∣B) or P(B∣A) |
| Formula | P(A) = ∑BP(A∩B) | P(A∣B) = P(A∩B)/P(B) |
| Example | Probability of rolling a 2 and flipping heads: P(2 ∩ Heads) | Probability of rolling a 2:P(2) | Probability of rolling a 2 given that the coin flip is heads: P(2 ∣ Heads) |
| Calculation Context | Calculated from a joint probability distribution. | Calculated by summing the joint probabilities over all outcomes of the other variable. | Calculated using the joint probability and the marginal probability of the given condition. |
| Dependencies | Involves multiple events happening simultaneously. | Does not depend on other events. | Depends on the occurrence of another event. |
| Use Case | Used to find the likelihood of combined events in probabilistic models. | Used to find the likelihood of a single event in the presence of multiple events. | Used to update the probability of an event based on new information. |
Read More,
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