In the previous segment, you understood how probability is defined formally and also learnt some of its properties. You were also introduced to the concept of ‘events’, situations or scenarios for which we compute the probabilities. In this segment, we will take a look at the different types of ‘events’ that can be defined. Note that earlier you considered a single event for which you.

computed the probabilities. Now, you will look at two or more events and understand how they are related to each other.

The two main categories of events that you need to know right now are **independent **events and **disjoint** or **mutually exclusive** events. Let’s learn their formal definitions.

**Independent events:**If you have two or more events and the occurrence of one event has no bearing whatsoever on the occurrence/s of the other event/s, then all the events are said to be independent of each other. For example, the chances of rain in Bengaluru on a particular day has no effect on the chances of rain in Mumbai 10 days later. Hence, these two events are independent of each other.

**Disjoint or mutually exclusive events:**Now, two or more events are mutually exclusive when they do not occur at the same time, i.e., when one event occurs, the rest of the events do not occur. For example, if a student has been assigned grade C for a particular subject in an exam, he or she cannot be awarded grade B for the same subject in the same exam. So, the events in which a student gets a grade of B or C for the same subject in the same exam are mutually exclusive or disjoint.

Now, what about events that are both independent and mutually exclusive? Do such events exist? Let’s hear from Amit as he explains the difference between the two in the upcoming video.

As explained in the video, two or more events cannot be independent and disjoint simultaneously. Two events can be either exclusively independent or exclusively disjoint. Here are some examples to drive home this point.

- The events ‘Customer A buys the product’ and ‘Customer B buys the product’ are independent, whereas the events ‘Customer A buys the product’ and ‘Customer A does not buy the product’ are disjoint.

The events ‘You will win Lottery A’ and ‘You will win Lottery B’ are independent, whereas ‘You will win Lottery A’ and ‘You will not win Lottery A’ are disjoint events.

**Complement Rule for Probability**

Now, disjoint events have one special property that is pretty intuitive and easy to understand. For example, let’s say A and B are two disjoint events. If A = ‘Event that it rains today’ and B =’ Event that it does not rain today’ and you know the P(A) = 0.3, can you guess what P(B) might be?

You must have guessed the answer here. It is nothing but 1 – P(A) = 1 – 0.3 = 0.7. This is something known as the** complement rule for probability**. It states that if A and A’ are two events which are mutually exclusive/disjoint and are complementary/in negation of each other (you can read **A’** as ‘**not A**‘), then:

** P(A) + P(A’) =1**

If the probability that a customer buys a product is 0.4, then the probability that he/she does not buy the product is 0.6.

- If the probability that you win the lottery is 33%, then the probability that you do not win the lottery is 67%, and so on.

- This rule is basically an extension from the basic rule of probability that you learnt previously –
**the sum of probabilities for all events always add up to 1**. You can read more about the complement rule for probability here.

Now, answer the following questions to strengthen your conceptual understanding of the aforementioned topics.

**Comprehension**

Box8 is an online food-ordering app that operates in four major cities of India – Mumbai, Bengaluru, Pune and Gurgaon. Next week, it is going to launch a month-long marketing campaign for its membership program –** Box8 Pass** – that gives access to a flat 25% discount on all orders and access to other exclusive discounts and benefits in the city of Mumbai.

Now, the company’s lead business analyst has estimated that there is a high chance that they will gain around 500 new subscribers at the end of this campaign. She also calculated that they need at least 200 new subscribers to make sure that they recuperate the costs of the campaign and break even.

Given this information, answer the following questions:

Now that you have learnt two of the most common types of events that you generally encounter – **independent** and **disjoint** events – you will learn some rules to compute the probabilities of those events using the addition and the multiplication rules.