 Hello and welcome to this session. In this session we shall learn designing and using simulations to find probability. Now in our earlier sessions we have learnt about experimental probability and theoretical probability. Now experimental probability is an estimate of the actual probability and theoretical probability is the likelihood of an event happening and experimental probability is equal to number of times event actually occurs upon total number of trials. Now suppose in an experiment a coil is tossed 10 times that is number of trials is equal to 10 and from these 10 trials head appears 6 times so experimental probability of getting a head is equal to 6 upon 10 which is equal to 0.6 and we know that theoretical probability of getting a head when we toss a coin is 1 upon 2 which is equal to 0.5 So experimental probability is an estimate of the actual probability and theoretical probability is the likelihood of an event happening. Now let us discuss how to perform simulations because a large number of trials is often required to obtain an accurate estimate it is sometimes helpful to use technology to simulate many trials of an experiment. Now simulations are performing actual experiments and then finding the probabilities of an event. Now we can perform simulations by using one or more objects like dye, coin, marbles, spinner etc. Now let us discuss an example and in this example it is given that Paula is better chess player than Lisa. She wins 2 by 3 of games played between herself and Lisa while Lisa wins only 1 by 3 of the games. Now suppose Paula and Lisa have to compete in a best of 10 game series in which a player who wins 6 games wins the series. What is the probability that Paula wins the series? Now here a dye will be used to simulate each game. Now since Paula wins 2 by 3 of games so a number 1, 2, 3 or 4 counts on rolling a dye then Paula wins that game. As here the probability of getting a number 1 or 2 or 3 or 4 on a dye is equal to 4 upon 6. That is number of favorable outcomes upon total number of outcomes and this is equal to 2 upon 3. So here we are getting the probability as 2 upon 3 which means if any of these numbers appears on a dye then Paula wins the game. And also it is given that Lisa wins 1 by 3 of games. So if a number 5 appears on a dye then Lisa wins that game. As here the probability P of getting a number 5 or 6 on a dye is equal to 2 upon 6 which is equal to 1 upon 3. So here we are getting the probability P as 1 upon 3 which means whenever the number 5 or 6 appears on the dye then Lisa wins that game. Now after forming the simulation we will find the probability of Paula winning the game by performing the simulation. For this we will follow the following steps. In the first step a roller dye and assign the roll as a win to either Paula or Lisa continue rolling until one player has won 6 times. And in the second step repeat the trial 100 times. And recall the results which we have obtained in a table like this. Now here in the table in the first column we have written the trials. In the second column we have written the rolls 10 times and in the third column we have written the winner. Now we know that if a number 1 2 3 or 4 comes on a dye then Paula wins the game. And if a number 5 or 6 comes on a dye then Lisa wins the game. Now in the first trial you can see a dye is rolled 10 times. And here these outcomes in favour of Paula it means a total of 6 outcomes are in favour of Paula. It means Paula wins in the first trial in the second trial also. Now in the third trial you can see that these outcomes are in favour of Lisa. That is these 6 outcomes are in favour of Lisa. It means Lisa wins in the third trial. And we will repeat the trial 100 times. Now in the next step we will estimate the probability that Paula will win the series. Now the total number of trials is 100. Now suppose in 100 trials Paula wins 70 games then probability that Paula wins is equal to 70 upon 100 which is equal to 0.70. Now this is our required answer. Now let us discuss some important remarks. Now when we say that Paula wins 2 by 3 of games then it means that expected probability or theoretical probability that Paula wins the series is 2 upon 3. That is why we chose 4 numbers from 6 on a dye in favour of Paula. And here the experimental probability is equal to 0.70. That is the experimental probability that Paula wins the series is 0.70. Now the theoretical or expected probability is equal to 2 upon 3 which is equal to 0.666 or 66.6%. And experimental probability is equal to 0.70 or 70%. Now here both the probabilities are very close to each other. And as the number of trials will increase it would get closer to expected value. So in this session we have learnt how to design a simulation and to draw results from them. And this completes our session. Hope you all have enjoyed the session.