# Conditional Probability: Definition and Key Concepts

Bonani Bose 9 Min Read

Conditional Probability may be explained as the likelihood of an event or outcome occurring based on the occurrence of a previous event or outcome. Usually, Conditional probability is calculated by multiplying the probability of the preceding event by the updated probability of the succeeding, or conditional, event.

My general observation says that in problems where the occurrence of one event affects the happening of the following event. These scenarios of probability are classic conditional probability examples.

In the context of mathematics, the probability of occurrence of any event A when another event B in relation to A has already occurred is known as conditional probability.

Our discussion would also include differences between Conditional and Unconditional Probability and round off with the basic differences between Conditional Probability and Joint Probability.

**What is Conditional Probability: Definition of Conditional Probability**

The conditional probability may be defined as the probability of one event occurring with some relationship to one or more other events.

It is to be noted that the conditional probability does not state that there is always a causal relationship between the two events, as well as it does not indicate that both events occur simultaneously.

The concept of conditional probability is primarily related to the Bayes’ theorem, which is one of the most influential theories in statistics.

The Formula for Conditional Probability may be explained as:

P(A|B) – the conditional probability; the probability of event A occurring given that event B has already occurred

P (A ∩ B) – the joint probability of events A and B; the probability that both events A and B occur at the same time

P(B) – the probability of event B

The formula above is applied to the calculation of the conditional probability of events that are neither independent nor mutually exclusive.

Experts on Conditional Probability suggest another way of calculating the Conditional Probability by using the Bayes’ theorem. The theorem can be used to determine the conditional probability of event A, given that the event B has occurred by knowing the conditional probability of event B, given the event A has occurred, as well as the individual probabilities of the event A and B. Mathematically, the Bayes’ theorem can be denoted in the following way:

**What is Conditional Probability: Conditional Probability for Independent Events**

Conditional Probability may be explained as two events that are independent of each other if the probability of the outcome of one event does not influence the probability of the outcome of another event. Therefore, the Conditional Probability of two independent events A and B may be represented as:

P(A|B) = P(A)

P(B|A) = P(B)

**What is Conditional Probability: Conditional Probability for Mutually Exclusive Events**

In probability theory, mutually exclusive events may be explained as the events that cannot occur simultaneously. In other words, if an event has already occurred, another event cannot occur. Thus, the conditional probability of the mutually exclusive events is always zero.

P(A|B) = 0

P(B|A) = 0

**What is Conditional Probability: Conditional Probability Examples**

### **Conditional Probability Examples** **using a table of data**

According to a research paper, a two-way table of data is one of the most common problems we see in Conditional Probability. Here, we take a look at how to find different probabilities using such a table.

Example

A survey asked full time and part time students how often they had visited the college’s tutoring center in the last month. The results are shown below.

In a survey conducted by a college both full time and part-time, students were asked how often they had visited the college’s tutoring center in the last two months. The results may be represented as follows:

Suppose that a surveyed student is randomly selected.

(a) What is the probability the student visited the tutoring center four or more times, given that the student is full time?

Conditional probability is all about focusing on the information you know. When calculating this probability, we are given that the student is full time. Therefore, we should only look at full-time students to find the probability.

(b) Suppose that a student is part-time. What is the probability that the student visited the tutoring center one or fewer times?

This one is a bit trickier, because of the wording. Let us put it in the following way:

Find: probability student visited the tutoring center one or fewer times

Assume or given: a student is part-time (“suppose that a student is part-time”)

Since we are assuming (or supposing) the student is part-time, we will only look at part-time students for this calculation.

(c) If the student visited the tutoring center four or more times, what is the probability he or she is a part-time student?

As stated above, we must make sure we know what is given, and what we are finding.

Find probability he or she is part-time

Assume or given: the student visited the tutoring center four or more times (“if the student visited the tutoring center four or more times…”)

For this question, we are only looking at students who visited the tutoring center four or more times.

**What is Conditional Probability: Difference between Conditional Probability & Joint**

**What is Joint Probability**

The joint probability may be explained as a measure of how likely it is for two (or more) things to occur. For instance, if you roll two dice, you have the probability of getting a six on the first and a four on the second. This is a classic example of Joint Probability, where the probability of occurrence of both results is possible.

**What is Conditional Probability**

Conditional probability, on the other hand, may be explained as a measure of how likely one situation is likely to happen if you are aware of the occurrence of another event.

For example, what is the probability that the second die shows a four if the sum of the numbers on the two dice is ten? If you know that the sum is ten, it turns out that it is far more likely that the second die is a four than if you knew nothing about the sum.

**What is Conditional Probability: Difference between Conditional & Unconditional Probability**

**Definition of Conditional Probability**

Conditional Probability may be explained as a probability that considers some other piece of information, knowledge, or evidence.

**Definition of Unconditional Probability**

Unconditional Probability may be explained as a probability that does not consider any other information, knowledge, or evidence.

Krishna Singh, an expert on mathematics and statistics, explains the difference between Conditional and Unconditional Probability with the following example:

**Conditional Probability Examples**

Pulling an ace out of a deck of cards and then drawing a second ace without replacing the first. You would have a 4/52 chance of getting the first ace, but a 4/51 chance (if you didn’t pull an ace) of the second, making the second conditional upon the results of the first.

**Unconditional Probability Examples**

Rolling a die. The fact that you got a 6 on one roll has no effect on whether you will roll a 6 later on.

**Data Science and Conditional Probability**

Data Science often uses statistical inferences to predict or analyze trends from data, while statistical inferences make use of probability distributions of data. Therefore, knowing probability and its applications are important to work effectively on data science problems.

Most data science techniques rely on Bayes’ theorem. Bayes’ theorem is a formula that describes at large how to update the probabilities of hypotheses when given evidence. You can build a learner using the Bayes’ theorem to predicts the probability of the response variable belonging to some class, given a new set of attributes.

Data Science is inextricably linked to Conditional Probability. Data Science professionals must have a thorough understanding of probability to solve complicated data science problems. A strong base in Probability and Conditional Probability is essential to fully understand and implement relevant algorithms for use.

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