Independent Events
Andrew is closely observing a juggler in the circus. He has a few colored juggling clubs  1 is red, 2 are green, and the remaining 3 are blue. All of a sudden, he makes a mistake. One of the juggling clubs is dropped and picked again. Now, the second juggling club is also dropped from his hand. What is the probability that the first juggling club is blue and the second juggling club is green? It can be quite confusing to answer. The clubs that Andrew might pick are not dependent on any previous incidents and thus the clubs could be any of the colors. Such events are called independent events and these are not affected by previous events. So, let's learn about these kinds of events.
In this article, we will learn about independent events and how to find the probability of both dependent events and independent events. Both these events are a part of probability and related in many ways. We also explore the difference between independent and dependent events and how to approach them. You can check out the solved examples to know more about the lesson and try your hand at solving a few interesting practice questions on independent events at the end of the page.
What Are Independent Events?
The two events are said to be independent events if the outcome of one event does not affect the outcome of another. Or, we can say that if one event does not influence the probability of another event, it is called an independent event. Independent events in probability reflect reallife events. For understanding this, we can take some examples like scoring good marks in an exam has no effect on what the neighbors are up to. Similarly, taking a cab to market has no effect on finding your favorite movie on Youtube. Another way of putting is that an independent event does not rely on another event to happen first.
What are the types of independent events? There are two types of events in probability which are often classified as dependent events or independent events. Let's study their difference.
Difference Between Independent Events and Dependent Events
The difference between independent events and dependent events is given in the table below.
Independent Events  Dependent Events 
1. Refers to the occurrence of one event not affecting the probability of another event.  1. Refers to the occurrence of one event affecting the probability of another event. 
2.Example: a) Riding a bike and watching your favorite movie on a laptop. 
2.Example: a) Not recharging your phone bill on time and having your call service suspended. b) Purchasing a few lottery tickets and winning the lottery. The more tickets we purchased, the greater our odds of winning. c) Suppose you have a box containing 10 green apples and 5 red apples. If we take out a red apple and ate it properly, we have a condition that we cannot replace it again with other apples that are in the box so without replacing it, the probability of getting another red apple in your second attempt is greatly changed because you took out a red apple the first time. Such events are said to be dependent events. 
3. Formula can be written as: P(A and B) = P(A)×P(B)P(A and B) = P(A)×P(B) 
3. Formula can be written as: P(B and A)=P(A)×P(B after A) 
Finding the Probability of Independent Events
For finding the probability of independent events we must go through with the formula of conditional probability which is given below: If the probability of events A and B is P(A) and P(B) respectively, then the conditional probability of event B such that event A has already occurred is P(A/B). The conditional probability formula is presented below.
\[ P\left( \dfrac AB \right)=\dfrac {P(A \cap B)}{P(B)} \text {or} \dfrac {P(B \cap A)}{P(B)}\]
Given, P(A) must be greater than 0. P(A) less than 0 means that A is an impossible event. In \(P(A \cap B)\), the intersection denotes the compound probability of an event.
Let's find the probability of independent events through an example in detail. Suppose, we have a box that contains 10 toys in which 7 toys are multicolored and 3 are blue. Based on this we know that the probability of drawing one multicolored toy is 7 over 10, or 0.7, and the probability of drawing a blue toy is 3 over 10, or 0.3
Method to Identify Independent Events
Before applying probability formulas, one needs to identify an independent event. Few steps for checking whether the probability belongs to a dependent or independent events:
Step 1: Check if it possible for the events to happen in any order? If yes, go to Step 2, or else go to Step 3
Step 2: Check if one event affects the outcome of the other event? If yes, go to step 4, or else go to Step 3
Step 3: The event is independent. Use the formula of independent events and get the answer.
Step 4: The event is dependent. Use the formula of dependent event and get the answer.
Related Topics
Tips and Tricks on Independent Events
You can use the belowgiven tips and tricks for solving problems on independent events.
 The probability of independent events occurring in sequence can be found by multiplying the results together.
 If the probability of one event does not affect the probability of another event, the events are independent.
 If the probability of one event affects the probability of another event, the events are dependent.
Solved Examples on Independent Events

Example 1: Joseph and David are playing with cards and in a pack, there are 52 cards. Joseph drew a card at random with a replacement. Then he asked David what is the probability of drawing a queen followed by a king?
Solution:
As we understand that this probability is having an independent event condition:
P (drawing a queen in the first condition) = 4/52
P (drawing a king in the second condition after a queen with replacement) = 4/52
P (drawing a queen followed by a king) = 4/52 × 4/52 = 16/2704 = 1/169Answer: P (drawing a queen followed by a king) = 1/169

Example 2: A juggler has seven red, five green, and four blue balls. During his stunt, he accidentally drops a ball and then picks it up. As he continues, another ball falls. What is the probability that the first ball that was dropped is blue, and the second ball is green?
Solution:As we know that the first ball is picked by the juggler, the size of the sample space for both balls is 16, because these events are independent.
The probability that the first ball is blue or P (blue ball) = 4/16
The probability that the second ball is green or P(green ball) = 4/16
The probability that the first ball is blue and the second ball is green:
P(blue and green)= P(blue) × P(green) = 4/16 × 4/16 = 1/16
Answer: Thus, the probability is 1/16 for both cases.

Example 3: In a survey, a company found that 6 out of 10 people eat pizza. If three people are chosen at random with replacement, what is the probability that all 3 people eat pizza?
Solution:
If three people are chosen at random with a replacement who eat pizza, then the probability that all 3 people eat pizza is:
P(person 1 likes pizza) = 9/10
P(person 2 likes pizza) = 9/10
P(person 3 likes pizza) = 9/10
P(person 1 and person 2 and person 3 like pizza) = 9/10 × 9/10 × 9/10 = 729/1000Answer: The probability that all 3 people eat pizza is 729/1000
FAQs on Independent Events
What Are Independent Events Examples?
Independent events are those events whose occurrence is not dependent on any other event. For example, if we flip a coin in the air and get the outcome as Head, then again if we flip the coin but this time we get the outcome as Tail. In both cases, the occurrence of both events is independent of each other.
How Do You Know if an Event Is an Independent Event?
Events A and B are independent if the equation P(A∩B) = P(A) · P(B) holds true. You can use this equation to check if events are independent; multiply the probabilities of the two events together to see if they equal the probability of them both happening together.
What Is the Rule for Independent Events?
If the probability of events A and B is P(A) and P(B) respectively, then the two events are independent if any of the following are true: P(AB)=P(A), P(BA)=P(B) and P(A and B)=P(A)⋅P(B)
How Do You Tell if an Event Is Independent or Dependent?
Two events A and B are said to be independent if the fact that one event has occurred does not affect the probability of occurrence of the other. Also if the occurrence of one event affects the probability of occurrence of the other event, then the two events are said to be dependent.
Can an Event Be Mutually Exclusive and Independent Event?
Mutually exclusive in math are a set of events that cannot happen at the same time. For example: when tossing a coin, the result can either be heads or tails but cannot be both. This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive.
Are Independent Events Disjoint?
Two disjoint events can never be independent, except in the case that one of the events is null. Events are considered disjoint if they never occur at the same time. For example, being a freshman and being a sophomore would be considered disjoint events. Independent events are unrelated events.
Do You Multiply Independent Events Probability?
In order to use the rule, we need to have the probabilities of each of the independent events. Given these events, the multiplication rule states the probability of occurrence of both events is found by multiplying the probabilities of each event.
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