 Hello and welcome to this session. In this session, first of all let us discuss discrete distribution and continuous distribution. Now if a variable can take an n value between the five values, continuous variable otherwise a discrete variable. Now some examples will clarify the difference between discrete and continuous variables. In the first example, let us suppose the fire department mandates that all firefighters must weigh between 150 and 250 pounds. The weight of a firefighter would be an example of a continuous variable since a firefighter's weight could take an n value between 150 and 250 pounds. In the second example, suppose we fill the coin and count the number of heads. The number of heads could be any integer value between 0 and plus infinity. However, it could not be any number between 0 and plus infinity. We could not, for example, get 2.5 heads. Therefore, the number of heads must be a discrete variable. Now let us discuss the binomial distribution. Now our binomial distribution is one of the discrete probability distribution. It is used when there are exactly two mutually exclusive outcomes of a trial and these outcomes are appropriately labeled success and failure. The binomial distribution is used to obtain the probability of observing r successes in n trials with the probability of success on a single trial denoted by p. And the probability occurrence of event E exactly in n trials is given by probability capital P of a random variable capital X is equal to r is equal to lc r into p raise to power r into 1 minus p vote raise to power n minus r. Where n is the number of events, n is the number of successful events, p denotes the probability of success on a single trial, 1 minus p denotes the probability of failure over n minus r whole factorial into r factorial. Now let us discuss an example. In this a coin is tossed, then what is the probability exactly 7 heads? Now let us start with its solution. It is given that a coin is tossed 12 times which means the number of trials is equal to 12 and we have to find what is the probability of getting exactly 7 heads? Now here the number of successful events that is r is equal to 7 since we define getting a head as success in the trial that is the probability of getting a head on anything that is p is equal to 1 by 2 which is equal to 0.5. Now by using this formula, we can find out the probability of occurrence of an event e exactly r times in as well as c r which is equal to n factorial over r factorial. So this is equal to 12 factorial the whole factorial which is equal to 12 factorial over 12 minus 7 that is into factorial. Now this is our 10 factorial tenderness 12 into 11 into 10 into 9 into 8 into 7 factorial all of them now 7 factorial will be cancelled with 7 factorial and this is equal to 11 into 10 into 9 into 8 all of them now 5 factorial is 5 into 4 into 3 into 2 into 1 now 2 into 4 is 8 3 into 3 is 9 so it will be cancelled with 4 and 5 into 2 is 10 so this is equal to 792 now p here is 0.57 so p raise to power r will be equal to 0.5 whole raise to power 7 which is equal to 0.007 8125 now this is equal to 1 minus 0.5 whole raise to power this is equal to 0.5 whole raise to power equal to 0.03125 we have discussed the formula earlier which is as the probability vector p of a random variable capital S is equal to r is equal to mc p raise to power r into 1 minus p whole raise to power and minus r now putting the values of mcr and 1 minus p 0.0078125 into 0.03125 which on calculating will be equal to 0.193319 bar graph of a binomial distribution now we have discussed earlier that is one of the discrete probability distribution and we can make histogram only for the continuous distribution so here we have plotted only bar graph for binomial distribution now the binomial distribution is used when a researcher is entrusted in the occurrence of an event for instance in a clinical trial a patient may survive or die number of survivors and not how long the patient survives after treatment distribution of observations for the binomial distribution with parameters n is equal to 10 and p is equal to 0.20 now let us discuss the first case when in 10 observations the preferred outcome is we have to do 0 where the number of successful events is 0 value visibility capital P of a random variable x which is equal to r is equal to mcr that is 10 c0 into p raised to power r that is 0.20 raised to power 0 into 1 minus p whole raised to power n minus r that is 1 minus 0.20 raised to power 10 minus 0 now on solving this is equal to now 10 c0 will be 1 0.20 whole raised to power 0 is also 1 and 1 minus 0.20 that is 0.10 is equal to 0.107 the preferred outcome in one time so here r is equal to 1 so the probability capital P of capital X is equal to r is equal to 10 c1 into 0.20 raised to power 1 into 1 minus 0.20 whole raised to power so this is equal to 10 minus 1 into 1 0.20 into 0.8 whole raised to power now this is equal to 10 factorial upon 9 factorial 1 factorial is 1 into 0.20 into 0.8 whole raised to power 9 now on calculating this will be equal to 0.268 we can find the probability of preferred outcome occurs 2 3 4 now let us see the bar graph for this binomial distribution now this is the bar graph for the binomial distribution which we have discussed now that is the binomial distribution with parameters n is equal to 10 and P is equal to 0.20 and here the first part shows that when in time observations preferred outcome occurs 0 times then the probability is equal to 0.107 and similarly the second part shows that when in time observations preferred outcome occurs only one time then the probability is 0.3 we have drawn the bars for the other cases also now we know that the binomial distribution is one of the discrete probability distribution so here we are plotting only the bar graph for the binomial distribution now an example that is used often to illustrate concepts of probability clearing is the tossing of a coin 0 1 3 outcomes are equivalent to 0 1 2 3 or 4 heads now the likelihood of obtaining 0 1 2 3 or 4 heads is respectively 1 by 16 16 1 by 6 in the event of tossing a coin the probability of getting a head is 1 by 2 and here the coin is tossed 4 times that is here n is equal to 4 to find the probability of obtaining 0 heads that is when r is equal to 0 then the probability capital P that is 4 raised to power 0 into 1 minus 1 by 2 that is 1 by 2 4 raised to power n minus r that is 4 minus 0 which is 4 4C naught is 1 1 by 2 whole raised to power 0 is 1 and 1 by 2 whole raised to power 4 is 1 by 60 so the likelihood of similarly the likelihood of obtaining 1 2 3 or 4 heads is respectively 4 by 16 4 by 16 and 1 by 6 now the other situations in which binomial distributions arise are market research public opinion surveys medical research and animal abundance problems so in this session you have learnt about the experience of continuous distributions binomial distribution and bar graph of the binomial distribution so this completes our session hope you all have enjoyed the session