Ever had the question of how the probability of an event will be affected by having prior knowledge about the event? Here comes the application of conditional probability. Conditional probability is used to determine the probability of an event given that another event has taken place.
For instance, in trying to determine the probability of a student passing in math tests after being made aware of the fact that the student had extra classes, then conditional probability is applied.
What Is Conditional Probability?
The term conditional probability denotes the chances of an event occurring on account of the other event happening beforehand.
Symbolically, conditional probability is expressed as P(A | B), which conveys that “the probability of A under the conditions of B.”
In the above expression:
- A denotes the events we are examining.
- B denotes the event or condition which has already happened.
- P(A | B) implies the probability of A occurring when B has happened.
It is worth noting that with regard to conditional probability, the available information modifies the circumstances of the case at hand.
Conditional Probability Calculations
Given the above definition, the calculation of conditional probability is shown by the following formula:
P(A | B) = P(A ∩ B) / P(B)
Where:
P(A | B) = likelihood of A occurring with knowledge of B
P(A ∩ B) = likelihood of A and B occurring together
P(B) = likelihood of B
The formula presupposes that P(B) does not equal 0.
Therefore, to calculate the conditional probability correctly, one needs to rely on the cases of the occurrence of event B, and see how often event A can happen at the same time.
Recommended reading: What are the Properties of Real Numbers?
How to Calculate Conditional Probability
You can calculate conditional probability using three simple steps:
- Identify the event you want to find.
- Identify the condition that is already known.
- Use the conditional probability formula.
Simple Example
Let’s take an example of a class containing 30 students, where 18 of them are into playing basketball, while out of those 18 students, 10 also play soccer.
If we know in advance that the student plays basketball, then what would be the probability of him or her playing soccer?
In this case:
- Event A = student plays soccer
- Event B = student plays basketball
- Total students playing both sports = 10
- Total students playing basketball = 18
Thus,
P(A|B) = 10/18 = 5/9
Therefore, P(A|B) = 5/9 or 55.56%.
It is clear from the above example that not all students (i.e., 30) are considered while calculating the probability.
Conditional Probability in Daily Life
The concept of conditional probability is not confined to academic studies only. Conditional probability finds application in various aspects of life and in the professional environment.
Conditional Probability in Real Life
- Weather forecast: Estimation of rain probability considering the atmospheric conditions.
- Medical testing: Estimation of the probability of disease for a person based on his/her test results.
- Sports: Estimation of the probability of winning of a team taking into account its present score.
- Business: Prediction of customers’ actions based on their previous experience.
- Insurance: Estimation of risks basing on the information about the customers and previous events.
- Machine learning: Prediction basing on the information.
It becomes clear why it is important to know about conditional probability.
Conditional Probability vs. Normal Probability
Conditional Probability vs. Normal Probability
Although they are related, normal probability and conditional probability are not exactly the same. The main difference is whether we use additional information about an event.
| Normal Probability | Conditional Probability |
|---|---|
| Finds the chance of an event occurring. | Finds the chance of an event under a known condition. |
| May not require additional information. | Uses additional information that is already known. |
| Example: Chance of rolling a 6 on a die. | Example: Chance of rolling a 6 given that the number rolled is even. |
