# Dependent and Independent Events

Dependent and Independent Events are the types of events that occur in probability. Suppose we have two events say Event A and Event B then if Event A and Event B are dependent events then the occurrence of one event is dependent on the occurrence of other events if they are independent events then the occurrence of one event does not affect the probability of other events.

We can learn about dependent and independent events with the help of examples such as the event of tossing two coins simultaneously the outcome of one coin does not affect the outcome of another coin then they are independent events. Suppose we take other experiments where we toss a coin only when we get a six in the throw of dice, where the outcome of one event is affected by other events then they are dependent events.

In this article, we will learn about dependent events, independent events, their formulas, examples, and others in detail.

## Dependent and Independent Events in Probability

An event in probability falls under two categories,

• Dependent Events
• Independent Events

Let’s learn about those in detail.

## Dependent Events

Dependent events are those events that are affected by the outcomes of events that had already occurred previously. i.e. Two or more events that depend on one another are known as dependent events. If one event is by chance changed, then another is likely to differ. Thus, If whether one event occurs does affect the probability that the other event will occur, then the two events are said to be dependent.

When the occurrence of one event affects the occurrence of another subsequent event, the two events are dependent events. The concept of dependent events gives rise to the concept of conditional probability which will be discussed in the article further.

### Examples of Dependent Events

For Example, let’s say three cards are to be drawn from a pack of cards. Then the probability of getting a king is highest when the first card is drawn, while the probability of getting a king would be less when the second card is drawn.

In the draw of the third card, this probability would be dependent upon the outcomes of the previous two cards. We can say that after drawing one card, there will be fewer cards available in the deck, therefore the probabilities after each drawn card changes.

## Independent Events

Independent events are those events whose occurrence is not dependent on any other event. If the probability of occurrence of an event A is not affected by the occurrence of another event B, then A and B are said to be independent events.

### Examples of Independent Events

• Tossing a Coin

Sample Space(S) in a Coin Toss = {H, T}

Both getting H and T are Independent Events

• Rolling a Die

Sample Space(S) in Rooling a Die = {1, 2, 3, 4, 5, 6}, all of these events are independent too.

Both of the above examples are simple events. Even compound events can be independent events. For example:

• Tossing a Coin and Rolling a Die

If we simultaneously toss a coin and roll a die then the probability of all the events is the same and all of the events are independent events,

Sample Space(S) of such experiment = {(1, H), (2, H), (3, H), (4, H), (5, H), (6, H), (1, T), (2, T), (3, T), (4, T) (5, T) (6, T)}.

These events are independent because only one can occur at a time and occurring of one event does not affect other events.

Note

• A and B are two events associated with the same random experiment, then A and B are known as independent events if

P(A ∩ B) = P(B).P(A)

## Difference Between Independent Events and Dependent Events

The difference between independent events and dependent events is discussed in the table below,

Independent Events

Dependent Events

Independent events are events that are not affected by the occurrence of other events.

Dependent events are events that are affected by the occurrence of other events.

The formula for the Independent Events is,

P(A and B) = P(A)×P(B)

The formula for the Dependent Events is,

P(B and A) = P(A)×P(B after A)

Examples of Independent Events are,

• Tossing one coin was not affected by the tossing of other coins
• Raining for a day and getting six in dice are independent events.

Examples of Dependent Events are,

• The probability of finding a red ball from a box of 4 red balls and 3 green balls changes if we take out two balls from the box.

### Mutually Exclusive Events

Two events A and B are said to be mutually exclusive events if they cannot occur at the same time. Mutually exclusive events never have an outcome in common. If we take two events A and B as mutually exclusive events where the probability of event A is P(A) and the probability of event B is P(B) then the probability of happening both events together is,

P(A∩B) = 0

Then the probability of occurring any one event is,

P(AUB) = P(A) or P(B) = P(A) + P(B)

## Conditional Probability Formula

Conditional probability formula tells the formula for the probability of the event when an event has already occurred. If the probability of events A and B are P(A) and P(B) respectively then the conditional probability of B such that A has already occurred is denoted as P(A/B).

If P(A) > 0, then the P(A/B) is calculated by using the formula,

P(A/B) = P(A ∩ B)/P(A)

In the case of P(A) = 0 means A is an impossible event, in this case, P(A/B) does not exist.

## Solved Examples on Dependent and Independent Events

Example 1: An instructor has a question bank with 300 Easy T/F, 200 Difficult T/F, 500 Easy MCQ, and 400 Difficult MCQ. If a question is selected randomly from the question bank, What is the probability that it is an easy question given that it is an MCQ?

Solution:

Total question in the question bank = 300 + 200 + 500 + 400

P(Easy) = (300+500)/1400 = 800/1400 = 4/7

P(MCQ) = (400+500)/1400 = 900/1400 = 9/14

P(Easy ∩ MCQ) = (500)/1400 =5/14

P(Easy|MCQ) = P(Easy ∩ MCQ)/P(Easy)

P(Easy|MCQ) = (5/14)/(9/14) = 5/9

Thus, the probability of an easy question given it is an MCQ is 5/9.

Example 2: In a shipment of 20 apples, 3 are rotten. 3 apples are randomly selected. What is the probability that all three are rotten if the first and second are not replaced?

Solution:

Total Apple = 20

Rotten Apple = 3

• Possibility of the first apple being rotten = 3/20
• Possibility of the second apple being rotten = 2/19
• Possibility of the third apple being rotten = 1/18

Probability of all three apples being rotten = P(3 Rotten) = (3/20 ×  2/19 × 1/18) = 6/6840 = 1/1140

Thus, the probability that all three apples are rotten is, 1/1140

Example 3: John has to select two students from a class of 10 girls and 15 boys. What is the probability that both students chosen are boys?

Solution:

Total number of students = 10 + 15 =25

Probability of choosing the first boy

P (Boy 1) = 15/25

P (Boy 2) = 14/24

P (Boy 1 and Boy 2) = P (Boy 1) and P (Boy 2)

P (Boy 1 and Boy 2) = (15/25) × (14/24) = 7/20

Thus, the probability of choosing both boys is 7/20

Example 4: A multiple-choice test consists of two problems. Problem 1 has 5 options and Problem 2 has 4 options. Each problem has only one correct answer. What is the probability of randomly guessing the correct answer to both problems?

Solution:

Here, the probability of the correct answer to Problem 1 = P(A) and the probability of the correct answer to Problem 2 = P(B) are independent events.

Thus the probability of a correct answer to Problem 1 and Problem 2 both = P(A ∩ B) = P(A). P(B)

• P(A) = 1/5
• P(B) = 1/4

P(A ∩ B) = (1/5) × (1/4) = 1/20

Thus, the probability of getting both answers correct is 1/20.

## FAQs on Dependent and Independent Events

### 1. What are Dependent Events in Probability?

Two events A and B are said to be independent events if the probability of occurrence of event A is not affected by the occurrence of another event B.

### 2. What are Independent Events in Probability?

Two events A and B are said to be dependent when the outcome of event A influences the outcome of event B.

### 3. What is the Formula to Find the Probability of Occurrence of A and B, when A and B are Dependent Events?

The formula for the probability of occurrence of A and B if A and B are dependent events is,

P(A and B) = P(A) · P(B|A)

### 4. What is the Formula to Find the Probability of Occurrence of A and B, when A and B are Independent Events?

The formula for the probability of occurrence of A and B if A and b are Independent events is,

P(A and B) = P(A) · P(B)

### 5. What is the Probability of an Impossible Event?

Probability of an Impossible Event is 0.

### 6. What is the Probability of a Sure Event?

Probability of a Sure Event is 1.

Whether you're preparing for your first job interview or aiming to upskill in this ever-evolving tech landscape, GeeksforGeeks Courses are your key to success. We provide top-quality content at affordable prices, all geared towards accelerating your growth in a time-bound manner. Join the millions we've already empowered, and we're here to do the same for you. Don't miss out - check it out now!

Previous
Next