 Hello and welcome to the session. In this session we shall learn the meaning of compound events and formula for finding probability of compound events that is independent and dependent events. Suppose we have one coin, if I flip the coin then what is the probability that head comes? Here I have one event of getting a head and when we have one event we call it simple event and probability of getting a head is given by favorable outcomes upon fatal outcomes. We know that when we flip a coin we either get a head or a tail. So the number of favorable outcomes of getting a head will be equal to 1 and total outcomes are given by 2. So we get the probability of getting a head as 1 upon 2, now suppose if I flip two coins then what is the probability of getting a head on one coin and a tail on the second? Here we have two events. Event A is equal to getting a head and event B is equal to getting a tail. So I have to find the probability P of getting a head on one coin and tail on the other coin. Such type of events are called compound events. Thus compound event is made up of two or more simple events. Some examples of compound events are getting a 3 or 4 in rolling of a dice. Getting a queen or a heart when a card is selected from a deck of 52 cards, getting two heads when a coin is tossed twice. Compound events are of various types such as independent events, dependent events, mutually exclusive events, mutually not exclusive events. Here we study independent events and dependent events. Two events are independent events. The occurrence of one event does not affect the likelihood that the other event will occur. Like in above example tossing of two coins is independent. The coming of head on one coin does not affect coming of tail on the other coin. So we say both events are independent of each other. If we toss two coins simultaneously we could get the following combinations that is head-head, tail-tail, tail-head, head-tail. So we have four outcomes and two are in favor of head on one coin and tail on the other coin. So probability of getting a head on one coin on the other coin is equal to favorable outcomes that is two and total outcomes given by four. So we have the probability as 1 by 2. We have a direct formula for finding the probability of independent events. That is if a and b two independent events then probability of a and b is given by probability of a into probability of b which can also be written as probability of a and b is equal to probability of a into probability of b here dot represents multiplication. Now using this formula we can find the probability of above example we have event a getting a head on first coin and event b is equal to getting a tail on second coin then probability of event a and event b is equal to probability of event a into probability of event b. Now probability of event a that is getting a head is equal to 1 by 2 and similarly probability of event b that is getting a tail is also equal to 1 by 2. Therefore probability of event a and event b is equal to probability of event a that is 1 by 2 into probability of event b that is 1 by 2 which is equal to 1 by 4. Now here there can also be cases when we get a tail on the first coin and a head on the second coin. So here let event b be given by getting a head on second coin and event b is given by getting a tail on first coin. So probability of event c and event d is given by probability of event c into probability of event d and probability of event c that is probability of getting a head is given by 1 upon 2 and similarly probability of event d that is probability of getting a tail is also equal to 1 by 2. Therefore we have probability of event c and event d is equal to probability of event c that is 1 by 2 into probability of event d that is 1 by 2. So we get this probability as 1 upon 4 and therefore total probability will be given by probability of event a and event b plus probability of event c and event d which is equal to 1 upon 4 plus 1 upon 4 that is 1 by 2. Now we are going to discuss dependent events. Two events are dependent events if the occurrence of one event does affect the likelihood that the other event will occur. Suppose I have 5 balls that is 3 red and 2 blue I randomly pick 1 ball and keep it aside. Now again I drop another ball. Now what is the probability that I choose red in first drop and blue in second. Now here we have two events. Event a that is drawing a red ball in first drop. Event b that is drawing a blue ball in second drop. 5 balls in all where there are two blue balls and three red balls. Now see that after drawing first ball I do not replace the drawn ball rather I kept it aside. I drew second ball without replacement which means when I drew second ball the total number of balls are reduced to 4 so first event affects the second event. So these two events are dependent events. Now probability of event a that is drawing a red ball in first drop is given by favorable outcomes by total outcomes. And here I have three red balls and I can draw any of the red balls so the number of favorable outcomes is equal to 3 and the total outcomes is given by 5. Now probability of event b after a will again be equal to favorable outcomes by total outcomes and for event b the total number of outcomes are reduced to 4. So we have total outcomes as 4 and the number of favorable outcomes will be equal to 2 as I have two blue balls and I can draw any of them. So probability of event a and event b is given by 3 upon 5 into 2 upon 4 which is equal to 3 by 10. We have a direct formula for finding probability of dependent events. If we have two events a and b which are dependent events then the probability of both events occurring is given by the product of probability of a and probability of b after a occurs to the above example probability of a was given by 3 by 5 and probability of b after a occurs was 2 by 4 that is probability of a is 3 by 5 and probability of b following a is 2 by 5 so probability of event a and b is equal to probability of event a that is 3 by 5 into probability of b following a that is 2 by 4 is equal to 3 by 10 which is the required answer. This completes our session hope you enjoyed this session.