Conditional Probability

Conditional Probability – Importance, Properties And Example

Probability is a way to check the change in one thing concerning the other one. Probability has so many theoretical as well as real-life implementations. One of the most commonly used types of probability is named conditional probability. Here you have to see what type of problem you have to solve. Based on the problem type, you have to go through further categories of conditional probability. Each category in conditional probability has great importance in the field of statistics. Following are some important aspects of conditional probability:

  • conditional probability and independence
  • conditional probability with Bayes’ theorem
  • conditional probability using a two-way table
  • conditional probability with a tree diagram

The aspects mentioned above related to conditional probability have different use in a particular sector. Let’s take one aspect as ‘conditional probability and independence’ and discuss it. In this probability, you have to see which two events do not depend on each other. For example, you have tossed a coin. Here you find results as a tail. If you link this situation with gender, does it make sense?

Similarly, when a girl tosses a coin, there will always be heads. In the same way, when a boy tosses a coin, there will be tails always. You can judge that these statements are illogical. Both of the events are not linked with each other. This shows that both events are independent of each other. Conditional probability is used in many business sectors. Hence in the light of its importance, this article aims to discuss different properties and the importance of conditional probability.

  1. What Is Conditional Probability Formula?

The formula of conditional probability is given below:

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

You can represent the same formula in another way like:

P (B|A) = P (A∩B) / P (A)

Let’s have a look at the parameter of the formula.


  • P is the probability for the happening of an event
  • A is the probability of Event A
  • B is the Event B that has been already occurred
  1. How Is Conditional Probability Used In Real Life?

Below is a real-life example of conditional probability as shared by experts of a coursework help service.

Suppose you have a business of AC (air conditioner) sales and purchases. You have to see how your sales and purchase face ups and downs. You can make two events over there. The first event is the sale and purchase of AC. On the other hand, the second event is seasonal changes. In summers, your sales’ and purchase rate increase up to a great extent. In the same way, you may have very few sales and purchases of AC in winter. Make a rough estimation in the form of a percentage. Here,

P (sale and purchase in winters) = 20%; P (sale and purchase in summer) = 80%.

That is how a business holder can find the probability of selling and purchasing different products.

  1. What Conditional Probability is and Why it is Important to Data Analysis?

Conditional probability can be used for data analysis. Its best example is in the management sector. At the management level, you may have to forecast many aspects. So, you can use conditional probability for future decisions. Based on the evaluated forecast, you can measure risk factors in your work. You can calculate the probability by making a logical sequence of events. The only thing that matters here is identifying events and accurate data for analysis. At the management level, you can also find uncertainty for some events. You can do so in different ways. The frequently used way is to use the record for that particular event. You can forecast further aspects using conditional probability based on the record.

  1. Conditional Probability Properties

Three properties are associated with conditional probability. These three properties are given below:

  • The first property if as

P (S|B) = P (B|B) = 1


S is the sample space

P is the probability

A and B are two events of probability.

  • The second property of conditional probability is represented as,

P ((A ∪ B)|F) = P (A|F) + P (B|F) – P ((A ∩ B)|F)

Here, you must be clear that P (F) ≠ 0


S is the sample space

P is the probability

A and B are two events of probability.

F is supposed to be the event of sample space.

  • The third and last property of conditional probability is given as:

P(A′|B) = 1 − P(A|B)


P is the probability

A and B are two events of probability.

Based on your problem type, you have to select one property and find solutions.

  1. Conditional Probability Examples And Solutions

Let’s make some examples and solve them through conditional probability.

  • Example Number 1:

Suppose you have two fruits. The one is orange, while the second fruit is mango. The probability of eating mango than an orange is 0.39. The probability of eating mango in the first phase is 0.4. You have to fund the probability of eating an orange in the second phase, given that you have eaten mango in the first phase.

Given data:

P (orange and mango) = 0.39

P (mango) = 0.4

To find:

P (orange|mango) =?


P (orange|mango) = P (orange and mango) / P(mango)


P (orange|mango) = 0.39 / 0.4

P (orange|mango) = 0.97

  • Example Number 2:

Harry took two tests. The first is diabetes. The second test is of uric acid. The probability of having positive results in both tests is 0.5. The probability of having a positive result on the diabetes test is 0.9. Find the probability of having positive uric acid results, given that he has a positive result in the diabetes test.

Given data:

P (diabetes and uric acid) = 0.5

P (diabetes) = 0.9

To find:

P (diabetes|uric acid) =?


P (diabetes|uric acid) = P(diabetes and uric acid) / P(diabetes)


P (diabetes|uric acid)) = 0.5 / 0.9

P (diabetes|uric acid) = 0.55

In the same way, you can find conditional probability in any case.

I hope you will find the above-discussed aspects beneficial.