Independence of Events

Independence of events or independent events states any two conditions that do not affect each other. In other words, we can regard the independent events in which one’s outcome does not affect the other’s outcome.

Imagine you want to get good marks in the exam. This outcome has nothing to do with your sister’s dance classes. Thus, now it is understood that any two unrelated events that don’t alter one another are independent events in mathematics. They are a vital part of the probability. 

How to Identify an Independent Event?

Let’s get to know how to identify an independent event. 

• Firstly, check if the events are happening in a certain specific order 
• If they are in a specific order, then check if the outcome of one event affects the other or not? If it does affect the outcome of other events, then they are dependent events
• If the events are not in any order, then they might be independent

The Formula for Finding Probability for Independent Events

If two events are independent, then you can use the formula given below to find the probability.

P (A/B) =P(A∩B)/ P(B)

Difference between Dependent and Independent Events

• Dependent events are any two such events whose outcome affects the other
• To simplify, if the outcome of one event alters or changes the outcome of the other, they are called dependent events
• On the other hand, two events that do not affect one another in any way are independent events

For example: If you purchase a lottery ticket, then you might win the lottery. You can see that purchasing a lottery ticket (event 1) affects the winning of it (event 2). The more lottery tickets you buy, the probability of winning increases.

For the independent event, assume you choose a 3 from the deck of cards. Now, you want to choose an ace after replacing it. As you can see, these two events are completely independent. Choosing an ace for the 2nd time after replacing 3 has nothing to do with choosing a 3 for the 1st time. 

Miscellaneous Questions on Independence of Events

Question 1- There is a box of glass balls in different colours. We count the balls and find there are 3 green balls, 4 red balls, and 2 blue balls. One of these balls is to be removed and then replaced by a kid. Now, a kid draws another ball from the box. What shall be the probability for the kid getting the blue ball at the first and the green ball at the second picking?

Solution-

Before drawing the second ball, the first ball was replaced. Therefore, it becomes an independent event, and there is no change in the total number (9) of balls too. 

P (blue than green)=P(blue)⋅P(green)

=2/27

Question 2- Elena and Caroline were playing cards at their favourite place. There are a total of 52 cards in the pack. Elena replaces her card with another one. Now, she asks her friend about the probability of drawing a queen followed by his king. Find the probability here. 

Solution-

Before drawing a card for the king or queen, Elena replaces the first card. Therefore, it is an independent event. 

Assume the probability as follows:

Probability of drawing a queen in the first condition = 4/52

 Probability of drawing a king in the second condition after a queen (with replacement) = 4/52

 Probability of drawing a queen followed by a king = 4/52 × 4/52 

= 16/2704 

= 1/169

Answer: Probability of drawing a queen followed by a king (P) = 1/169

Conclusion 

Independence of events or independent events refers to any two conditions that do not affect each other’s outcomes. Whereas, Dependent events are any two such events whose outcome affects the other. An example of an independent event shall be the tossing of the coin. As we tossed the coin,  the chances of getting a head in the first flip does not affect the chance of getting a tail in the second flip. An example of a dependent event is waking up early and attending an early morning meeting. Both events depend upon one another. They have great importance in the field of probability in mathematics.