 Hello and welcome to this session. In this session we will discuss additional general for mutually inclusive and exclusive events. First of all let us discuss mutually inclusive events. Now let us consider a sample page of tossing 3 coins. Now here you can see page of tossing 3 coins. Now here F3 presents head and T represents tail. Now whenever you toss 3 coins these are all possible outcomes that is getting all heads. When getting head on first coin getting head on second coin as getting a tail on third coin these possible outcomes. So sample space S is a set containing all these elements. Now here this is equal to 1, 2, 3, 4, 5, 6, 7 and 8. So total number of outcomes is equal to 8. Now let us consider two events. Event A is getting exactly one head and event B is getting at least two tails. Now let us write elements for event A. Now event A is getting exactly one head. Now here these are the outcomes available to event that is where we are getting exactly one head. Now event A is a set containing elements T, T, H, T, H, T and H, T, T. So number of favorable outcomes for event A is equal to we can write elements of event B. Now event B is getting at least two tails. Now here you can see these are the outcomes favorable to event B. So event B is a set containing elements T, T, H, T, H, T, H, T and T, T, T. So number of favorable outcomes for event B is equal to here. Now let us see outcomes that are favorable to both the events A and B. Now there are three outcomes that are common to both the events. So A intersection B that is event A intersection B is a set containing elements T, T, H, T and H, T, T. Now number of favorable outcomes for event A intersection B is equal to 3. When the two events have some common outcomes then they are called usually inclusive events. Now see in the brain diagram there is a common area in both the events. So these events are inclusive events. They are also called not mutually exclusive events. So here A and B mutually inclusive events. Now we have to find probability of event A or B. This is written as P of A or B. Or we can write it as P of A union B. Now you find probability P of event A or B. We make use of addition rule that is probability P of event A union B is equal to probability of event A plus probability of event B minus probability of event A intersection B or we can write it as probability P of event A or B is equal to probability of event A plus probability of event B. Minus probability of event A and B. Now let us find probability of event A or B for above example. Now here total number of outcomes is equal to 8. And number of favorable outcomes for event A is equal to 3. P of event A is equal to number of favorable outcomes for event A that is 3 upon total number of outcomes that is 8. So probability of event A is 3 upon 8. Similarly probability of event B is equal to 4 upon 8 which is equal to 1 upon 2 probability of event A intersection B is equal to 3 upon 8. Now we know that probability of event A union B is equal to probability of event A plus probability of event B minus probability of event A intersection B. So for above example probability of event A union B will be equal to probability of event A that is 3 upon 8 plus probability of event B that is 1 upon 2 minus probability of event A intersection B that is 3 upon 8 and this is further equal to 1 upon 2. So probability of event A union B and probability of event A or B is equal to 1 upon 2. So using addition rule we have found probability of event A or B. Now let us discuss mutually exclusive events. Now mutually exclusive events are disjoint events. That is the two events will have no common answer. Now see this wing diagram. Here the two sets have no common area. So A intersection B is empty set. Now for mutually exclusive events probability of event A intersection B is equal to 0. Now addition rule for mutually exclusive events is given by probability P of event A or B is equal to probability of event A plus probability of event B. For example in dozen of three coins if we take event A as getting exactly 3 hats then event A is a set containing single element h h h. The number of available outcomes for event A is equal to 1. So here probability of event A will be equal to 1 upon 8 and event B is getting at least 2 tails. Now here you can see these are the outcomes available to event B. So event B is a set containing elements t t h t h t h t t and t t t. Number of available outcomes for event B is equal to 7. So probability P of event B is equal to 4 upon 8 which is equal to 1 upon 2. Now this is event A and this is event B. And here you can see the two events have no common element. So A intersection B is equal to empty set that is denoted by 5. Therefore probability of event A intersection B is equal to 0. So probability P of event A union B is equal to probability of event A plus probability of event B. Now probability of event A is 1 upon 8 and probability of event B is 1 upon 2. So probability of event A union B is equal to 1 upon 8 plus 1 upon 2. Now taking LCM of denominators that is LCM of 8 and 2 is equal to 8. So in denominator we have 8. Now in numerator we will have now here we know that 8 into 1 is 8 and 1 into 1 is 1. Plus 2 into 4 is 8 and 4 into 1 is 4. So this is equal to 5 upon 8. So probability of event A union B is equal to 5 upon 8. So in this fashion we have discussed how to find probabilities of mutually inclusive and exclusive events using addition rule. And this completes our session. Hope you all have enjoyed the session.