Conditional Probability

Conditional probability is one of the types of probability in probability theory, where the probability of one event is dependent on the other event already happened. As this type of event is very common in real life, conditional probability is often used to determine the probability of such cases.

In this article, we will learn about the topic of “conditional probability” in detail, including its mathematical definition, formula, conditions of dependent and independent events based on the formula, and many more. So, let’s start our learning of this fundamental topic “Conditional Probability”.

What is Conditional Probability?

Conditional probability is the probability that depends on a previous result or event. Due to this fact, they help us understand how events are related to each other. Simply put, conditional probability tells us the likelihood of the occurrence of an event based on the occurrence of some previous outcome. 

With the help of conditional probability, we can tell apart dependent and independent events. When the probability of one event happening doesn’t influence the probability of any other event, then events are called independent, otherwise dependent events.

Learn more about dependent and independent events.

Definition of Conditional Probability

Conditional Probability is defined as the probability of any event occurring when another event has already occurred. 

In other words, it calculates the probability of one event happening given that a certain condition is satisfied. It is represented as P(A | B) which means the probability of A when B has already happened.

For Example, let’s consider the case of rolling two dice, sample space of this event is as follows:

{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),
(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),
(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),
(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),
(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),
(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

Now, consider an event A = getting 3 on the first die and B = getting a sum of 9.

Then the probability of getting 9 when on the first die it’s already 3 is P(B | A),

which can be calculated as follows:

All the cases for the first die as 3 are (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6).

In all of these cases, only one case has a sum of 9.

Thus, P(B | A) = 1/36.

In case, we have to find P(A | B),

All cases where the sum is 9 are (3, 6), (4, 5), (5, 4), and (6, 3).

In all of these cases, only one case has 3 on the first die i.e., (3, 6)

Thus, P(A | B) = 1/36.

Conditional Probability Formula

As we can calculate the Conditional Probability of simple cases without any formula, just as we have seen in the above heading but for complex cases, we need a formula as we can’t possibly count all the cases for those. Let’s consider two events A and B, then the formula for conditional probability of A when B has already occurred is given by:

P(A|B) = P(A ∩ B) / P(B)

Where,

• P(A ∩ B) represents the probability of both events A and B occurring simultaneously, and 
• P(B) represents the probability of event B occurring. 

In other words, the conditional probability of A given B has already occurred is equal to the probability of the intersection of A and B divided by the probability of event B.

How to Calculate Conditional Probability?

To calculate the conditional probability, we can use the following step-by-step method:

Step 1: Identify the Events. Let’s call them Event A and Event B.

Step 2: Determine the Probability of Event A i.e., P(A)

Step 3: Determine the Probability of Event B i.e., P(B)

Step 4: Determine the Probability of Event A and B i.e., P(A∩B).

Step 5: Apply the Conditional Probability Formula and calculate the required probability.

Conditional Probability of Independent Events

When two events are independent, those conditional probability is the same as the probability of the event individually i.e., P(A | B ) is the same as P(A) as there is no effect of event B on the probability of event A. For independent events, A and B, the conditional probability of A and B with respect to each other is given as follows:

• P(B|A)= P(B)
• P(A|B)= P(A)

Conditional Probability Vs Joint Probability Vs Marginal Probability

The difference between Conditional Probability, Joint Probability, and Marginal Probability is given in the following table:

Parameter	Conditional Probability	Joint Probability	Marginal Probability
Definition	The probability of an event occurring given that another event has already occurred.	The probability of two or more  events occurring simultaneously.	The probability of an event occurring  without considering any other events.
Calculation	P(A | B)	P(A ∩ B)	P(A)
Variables involved	Two or more events	Two or more events	Single event

Conditional Probability with Example

There are various examples of conditional probability as in real life all the events are related to each other and happening any event affects the probability of another event. For example, if it rains, the probability of road accidents increases as roads have less friction. Let’s consider some problem-based examples here:

Tossing a Coin:

Let’s consider two events in tossing two coins be,

A: Getting a head on the first coin.

B: Getting a head on the second coin.

Sample space for tossing two coins is:

S = {HH, HT, TH, TT}

The conditional probability of getting a head on the second coin (B) given that we got a head on the first coin (A) is = P(B|A)

Since the coins are independent (one coin’s outcome does not affect the other), P(B|A) = P(B) = 0.5 (50%), which is the probability of getting a head on a single coin toss.

Drawing Cards:

In a deck of 52 cards where two cards are being drawn, then let’s consider the events be

A: Drawing a red card on the first draw, and

B: Drawing a red card on the second draw.

The conditional probability of drawing a red card on the second draw (B) given that we drew a red card on the first draw (A) is = P(B|A) 

After drawing a red card on the first draw, there are 25 red cards and 51 cards remaining in the deck. So, P(B|A) = 25/51 ≈ 0.49 (approximately 49%).

Properties of Conditional Property 

Some of the common properties of conditional property are:

Property 1: Let’s consider an event A in any sample space S of an experiment.

P(S|A) = P(A|A) = 1

Property 2: For any two events A and B of a sample space S, and an event X such that P(X) ≠ 0,

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

Property 3: The order of set or events is important in conditional probability, i.e.,

P(A|B) ≠ P(B|A)

Property 4: The complement formula for probability only holds conditional probability if it is given in the context of the first argument in conditional probability i.e.,

P(A’|B)=1-P(A|B)

P(A|B’) ≠ 1-P(A|B)

Property 5: For any two or three independent events, the intersection of events can be calculated using the following formula:

• For the intersection of two events A and B,

P(A ⋂ B) = P(A) P(B)

• For the intersection of three events A, B, and C,

P(A ⋂ B ⋂ C) = P(A) P(B) P(C)

Read More,

• Types of Events in Probability
• Mutually Exclusive Events
• Coin Toss Probability Formula

Conditional Probability Examples and Solutions

Example 1: A bag contains 5 red balls and 7 blue balls. Two balls are drawn without replacement. What is the probability that the second ball drawn is red, given that the first ball drawn was red?

Solution:

Let the events be,

Event A: The first ball drawn is red.

Event B: The second ball drawn is red.

P(A) = 5/12 

and P(B) = 4/11 (as first ball drawn is already red, thus only 4 red balls remain in the bag)

Therefore, the probability of the second ball drawn being red given that the first ball drawn was red is 4/11.

Example 2: A box contains 5 green balls and 3 yellow balls. Two balls are drawn without replacement. What is the probability that both balls are green?

Solution:

Let events be:

Event A: The first ball drawn is green, and

Event B: The second ball drawn is green.

P(A) = 5/8

P(B) = 4/7 ( as there are 4 green balls left out of 7)

Thus, the probability that both balls drawn are green is (5/8) × (4/7) = 20/56 = 5/14.

Example 3: In a bag, there are 8 red marbles, 4 blue marbles, and 3 green marbles. If one marble is randomly drawn, what is the probability that it is not blue?

Solution:

Let the events be:

Event A: The marble drawn is not blue, and 

Event B: The marble drawn is blue.

As A and B are complementary Events, we know

P(A) = 1 – P(B)

⇒ P(A) = 1 – 4/15 

⇒ P(A) = (15 – 4)/15 

⇒ P(A) = 11/15

Thus, probability of drawing a marble out of bag which is not blue is 11/15.

Example 4: In a survey among a group of students, 70% play football, 60% play basketball, and 40% play both sports. If a student is chosen at random and it is known that the student plays basketball, what is the probability that the student also plays football?

Solution:

Let’s assume there are 100 students in the survey.

Number of students who play football = n(A) = 70

Number of students who play basketball = n(B) = 60

Number of students who play both sports = n(A ∩ B) = 40

To find the probability that a student plays football given that they play basketball, we use the conditional probability formula:

P(A|B) = n(A ∩ B) / n(B)

Substituting the values, we get:

P(A|B) = 40 / 60 = 2/3

Therefore, the probability that a randomly chosen student who plays basketball also plays football is 2/3.

Example 5: In a deck of 52 playing cards, 4 cards are drawn without replacement. What is the probability that all 4 cards are aces, given that the first card drawn is an ace?

Solution:

Let the events be,

Event A: The first card drawn is an ace,

Event B: The second card drawn is an ace,

Event C: The third card drawn is an ace, and

Event D: The fourth card drawn is an ace.

P(A) = 4/52 (there are 4 ace out of 52)

P(B | A) = 3/51 (one is already drawn, thus 3 ace left)

P(C | A and B) = 2/50 (two is already drawn, thus 2 ace left)

P(D | A and B and C) = 1/49 (three is already drawn, thus 1 ace left)

To find the probability that all four cards are aces, we multiply the probabilities of the individual events.

P(A and B and C and D) = P(A) × P(B|A) × P(C|A and B) × P(D|A and B and C)

= (4/52) × (3/51) × (2/50) × (1/49)

= 1/270725

Therefore, the probability that all 4 cards drawn are aces, given that the first card drawn is an ace, is 1/270725.

Applications

Finance and Risk Management

Example: Assessing the probability of default for a borrower given certain financial indicators.

Application: Banks and financial institutions use conditional probability to evaluate the risk associated with loans and investments.

Healthcare and Diagnostics

Example: Determining the probability of a patient having a specific disease given the results of diagnostic tests.

Application: Conditional probability is crucial in medical diagnoses and decision-making, helping healthcare professionals make informed decisions based on test results.

Marketing and Customer Relationship Management (CRM)

Example: Predicting the probability of a customer making a purchase based on their past buying behavior.

Application: Businesses use conditional probability to tailor marketing strategies, optimize customer experiences, and personalize product recommendations.

Machine Learning and Artificial Intelligence

Example: Predicting the likelihood of a user clicking on a particular ad based on their online behavior.

Application: Conditional probability is fundamental in machine learning algorithms for tasks such as classification, recommendation systems, and natural language processing.

Weather Forecasting

Example: Estimating the probability of rain tomorrow given today’s weather conditions.

Application: Meteorologists use conditional probability to make weather predictions based on historical data and current atmospheric conditions

FAQs on Conditional Probability

1. What is Conditional Probability?

Answer:

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred or is known to have occurred. It is denoted as P(A|B), which reads as “the probability of event A given event B.” 

2. How is Conditional Probability Calculated?

Answer:

Conditional probability is calculated using the formula: 

P(A|B) = P(A ∩ B) / P(B)

Where,

• P(A ∩ B) represents the probability of both events A and B occurring simultaneously, and 
• P(B) represents the probability of event B occurring. 

3. What is the Difference between Conditional Probability and Regular Probability?

Answer:

Regular probability, often referred to as unconditional probability, calculates the likelihood of an event occurring without any prior information and is the basic probability of an event in isolation. On the other hand, conditional probability takes into account additional information or the occurrence of another event to calculate the probability of a particular event.

4. Can Conditional Probability be Greater than 1?

Answer:

No, conditional probability cannot be greater than 1 as the conditional probability is a type of probability. It can only be between 0 and 1.

5. What is the Relationship between Conditional Probability and Independence?

Answer:

Two events A and B are considered independent if the occurrence of one event does not affect the probability of the other event.

In terms of conditional probability, if events A and B are independent, then P(A|B) = P(A), and similarly, P(B|A) = P(B). In other words, the conditional probability of A given B is equal to the unconditional probability of A, and vice versa.

6. Can Conditional Probability be Negative?

Answer:

No, conditional probability cannot be negative as probabilities are always non-negative values between 0 and 1.

7. What is Conditional Probability with an example?

Answer:

Conditional probability, denoted P(A|B), assesses the likelihood of event A happening, given that event B has already occurred. For example, in a medical test scenario, it calculates the chance of having a disease when the test result is positive. It’s a key concept in probability theory, helping us understand real-world situations where one event depends on another.

8. What is Conditional Probability Class 10?

Answer:

In class 10, conditional probability is about finding the chance of one event happening given another has occurred. For example, it helps determine the probability of rolling an even number when you already know the number rolled is greater than 3. It’s a basic concept in probability used to solve real-world problems.

9. What are the Applications of Conditional Probability?

Answer:

Conditional probability finds practical use in:

1. Medical Diagnosis: Assessing disease probabilities from symptoms.

2. Weather Forecasting: Predicting weather conditions based on data.

3. Finance and Insurance: Analyzing risks and setting premiums.

4. Quality Control: Detecting defects in manufacturing.

5. Machine Learning: For recommendations and predictions.

6. Epidemiology: Studying disease spread and prevention.

7. Speech Recognition: Predicting words in language.

8. Game Theory: Analyzing strategic decisions.

9. Genetics: Assessing genetic trait inheritance.

10. Sports Analytics: Predicting outcomes in sports.

11. Marketing: Understanding customer behavior.

12. Risk Management: Evaluating and mitigating risks.

It’s a versatile tool for decision-making and prediction in many fields.

10. What are the Types of Conditional Probability?

Answer:

1. Simple Conditional Probability: Probability of one event given another.

2. Joint Probability: Probability of multiple events occurring together.

3. Marginal Probability: Probability of a single event without conditions.

4. Conditional Probability Distribution: Probability distribution based on conditions.

5. Posterior Probability: Updated probability after considering new evidence.

6. Prior Probability: Initial probability before new evidence.

7. Sequential Conditional Probability: Probability of events occurring in a specific sequence.

8. Time Series Analysis: Predicting future events based on past data.

9. Markov Chains: Transition probabilities between states in a sequence.