Independent Events

In terms of probability, two events are independent if the occurrence of one event does not affect the likelihood of the other. Determining the independence of events is important because it provides whether the multiplication rule should be applied to the probability calculation. If the events you’re working on are independent, calculating probabilities using product rules is fairly easy. Calculating the probability of a dependent event can be more complex and less straightforward. Therefore, it is important to think about whether the events are independent or not. Because it affects your approach to problem-solving. For collections of three or more events, it is necessary to distinguish between the notions of weak independence and those of strong independence. When any two events in a collection are independent of each other,  the events are said to be independent of each pair, but when the events are independent of each other (or collectively independent). Intuitively means that each event is independent of the combination of other events. A collection of random variables has a similar concept. 

 The name “mutual independence” (such as “collective independence”) seems to be the result of an educational choice to distinguish between the strong and weak terms “independence in pairs”. .. In the advanced literature of probability theory, statistics, and stochastic processes, the stronger term is simply called unqualified independence. Independence is more powerful because it means independence for each pair, but not the other way around.

Events A and B are said to be independent if the probability of occurrence of event A does not depend on the occurrence of other events B. Consider an example of a coin flip. 

 P(A) = 3/6 = ½, P(B) = 2/6 = ⅓ if A is an event “the number  is odd” and B is “the number comes up is a multiple of 3” 

 Also, since A and B are events where ‘the number that appears is an odd number and is  a multiple of 3’, 

 P(A ∩ B) = ⅙ 

 P(A│B) = P(A ∩ B) / P(B) =  ⅙  /  ⅓  =  ½

 P (A) = P (A│B) = ½ ,

       which means that the occurrence of event B did not affect the probability of occurrence of event A. 

 If A and B are independent events, then P( A│B) = P(A) 

 Using the probability rule of multiplication P(AB) = P(B) P(A│B) 

 Note: A and B are two events related to the same randomized trial. If P(A ∩ B) = P(B), then A and B are known to be independent events. P(A) 

P(A∩B) = P(B) P(A)

How do you know the difference between a dependent and an independent event? 

The ability to distinguish between dependent and independent events is important for solving stochastic questions. Why? Imagine one event where you win the lottery. It depends on whether you are buying a ticket. Therefore, winning the lottery and buying tickets are dependent events. When you buy a ticket, the chances of winning a lottery can be 1/1 million. But what about other things, such as commuting by car or winning the lottery? If you drive a car (and don’t buy a ticket), you have no chance of winning the lottery. Therefore, the odds vary greatly depending on the event type. It can be difficult to know whether an event is a dependent or independent event. Not all situations are as simple as they seem. 

Types of Independent Events

• Conditional Independent Event – The concept of conditional probability is closely related to the concept of independent events. Some of the previous examples can be reformulated using conditional probabilities.  

Two events R and S are independent if:

P(R\S) = P(R\S`) and P(S\R)=P(S/R’) 

• Mutual Independent Event –  Given a set of three or more events, the sets of events are independent of each other, provided that each event does not depend on all the intersections of the other events. 

Two identical hexahedral dice (red and blue) are tossed. Let R be the event when the red die rolls 3, S be the event when the blue die rolls 4, and let T be when the total number of dice is 7. Are R, S, and T  independent?

P ( R/S ) = ⅙ and P ( R/S’ ) =  ⅙ . Thus R and S are independent.

P ( R/T ) = ⅙ and P ( R/T’ ) = ⅙ . Thus R and T are independent. 

P ( S/T ) = ⅙  and P ( S/T’ ) = ⅙ . Thus S and T are independent. 

These events are pair-independent. However, for all three events to be mutually independent, each event must be independent with all intersections of the other events.

Card Example of an Independent Event: Maps are often used to explain how seemingly independent events can affect other events. For example, if you choose a card from a pile of 52 cards, your chance of winning a jack is 4 out of 52. Mathematically, we can write: P(jacks) = number of jacks in  deck of cards / total number of cards in  deck = 4/52 = 1/13 ≈ 7.69%. If you change the jack and reselect (assuming the cards are shuffled), the event is independent. Your odds remain the same (1/13). Continuing to choose cards will be an independent event. Because every time you pick a card (a “test” of your odds), it’s a separate, unrelated event. 

Coin Example of an Independent Events: Toss a coin in the air and if the result is heading, flip the coin again, but this time you get the result with the backside. 

Dice Example of an Independent Events: The win 6 event on the first die roll and the win 6 event on the second die are independent. In contrast, an event that rolls a 6 on the first die and adds up to 8 on the first and second attempts is not independent.

Conclusion

Two events are said to be independent if the occurrence of one event does not affect the probability of the second event. Some examples of independent events are when two coins are tossed the first coin will show the head and the second coin will show tails are independent events. For proving three or more events independent every event has to be independent of the intersection of other events.