Hello, and welcome to the fourth exercise of this online course. During the Rio Olympic Games in 2016, Simone Biles did a fantastic performance and won five medals, four of them of gold. She conducted such a fantastic performance that one of her special moves was granted her name to honor them. This is a Biles Move. Well, in the Olympic games despite of all the fun of watching and all the Olympic spirit and nice spirit of the game, there is a lot of statistics behind a lot of people. But on the different disciplines a lot of people tend to see the games of some kind of probabilities of winning of losing. And today we are going to use some information about the Olympic game games specially about Simone Biles, this fantastic artistic gymnastic to have our exercise today and answer some specific questions about probability. So if we go to the materials that we have here. We can assume or we know that Simone Biles this gold medalist can conduct his perfect Biles Move In 95% of the competitions. Meaning 95% of the times that she conducts a Biles Move in a competition, she does it perfectly. We can now dive into the questions. First question asked is, consider the statistics given about Biles in this exercise and assume that Biles conducting a perfect Biles is a Bernoulli random variable. This means that there's only success or failure. Either she does the Biles Move perfect or she fails. The question now is what's the expected value of a perfect Biles Move? Well, since we are now moving or we are considering a Bernoulli random variable, the probability of a perfect Biles Move is 95% and represents the expected value of saying a perfect Biles Move. Now we can move to the second question. What's the variance and standard deviations of the perfect Biles Move according to the statistics? Well, we know from the theory that you learned from CAL that the formula for the variance in a Bernoulli random variable, is the probability multiplied by one minus the probability of the test. In this case we will take the probability of seeing a perfect Biles Move which is 0.95 and we will multiply it by 0.05 following these rules. Hence, this 0.0475 is the result of the variance. Similarly we can apply the rule to calculate standard deviatioin of this Bernoulli random variable which is just to take the square root of the variance. If we take the square root of the variance as we can do it here we just receive a value of 0.21794, this is a standard deviation. Now we are asked to use the probability mass function to calculate the probability that if Biles competes 35 times in the next years, she conducts at least 33 perfect Biles Moves. Okay, we have the hint here that we need to use the BINOM.DIST command in command in Excel. And we need to interpret this formula because this formula yields the probability of a certain event to happen at most n times, this means less than n times. Hence, if we want to calculate the probability of seeing at least 33 perfect Biles, if Simone Biles compete 35 time we need to calculate the probability of saying at most two failure to Simone Biles Moves out of 35 moves. The probability of seeing a failed Simone Biles Move is 0.05, meaning the difference between one and the probability of seeing a perfect Biles Move which was 0.95. And we need to here write true to state that the formula is not cumulative meaning that we are looking for at most two failures. Well, if we conduct this calculation, we see that the probability that we see at least 33 perfect Biles if Simone Biles competes 35 times is of, rounded, 75%. Now we have the last question which is use the same probability mass function to calculate the probability that if Biles competes again, 35 times in the next year. She conducts exactly 33 perfect Biles Move. Well, now we can calculate something the same formula two failures out of 35, probability of failure 0.05. And in this case we would just right here false to say that we are looking for exactly 33 perfect Biles. This would get a probability of 27.4, roughly. If we would like to apply the same formula, but using different data, meaning using a different form we could also use this. Look what I did here below, is instead of calculating the probability of two failures, I would calculate the probability of 33 successes out of 35. So we could come up here and would say probably 33 perfect Biles Moves out of 35. Thrice, attempted to dance with the probability of success of 95%. Again, false to state the we're looking for the exact event, and we will get the exact same probability. Thank you very much, I hope that you enjoyed exercise four, and see you in the next and last exercise. Have fun.