9 6 Simulate Chance Events Only Part

today we started on the very last lesson of module 10 this is going to be lesson six we're gonna be talking about simulated chances you can see the suggested pacing guide says that this is gonna be two day lesson there's only nine resources in it so don't know about that we shall have to see where it takes us to today it's pretty easy to roll number cube once or twice what if you had to roll the number cube 1 000 times would that take long time it absolutely would in fact that's not really something would want to have to do and have to document all those answers as you go when an experiment is difficult or time consuming to perform in real life you could use simulation to get results to simulate many trials of probability experiment computer can be used to generate results quickly in other cases when studying real life situations you can use number cubes coins or spinners in order to run these simulations how can random number generator help you to simulate rolling number cube 1000 times well hopefully we're going to learn about that in today's lesson our inquiry question says how could you use random number generator to model the probability of an experiment and they're going to let us use web sketch pad to do this the random number generators can be adjusted so the range of the numbers can accommodate any simulations this sketch will randomly generate numbers 1 through 6 to simulate rolling number cube select roll number cube 10 time to conduct simulation of 10 rolls and when you do boom there they are so there was simulation of 10 rolls of the number cube and if you see we reset that every time we do it we're going to get different numbers and see how quickly those numbers come in whole lot quicker than if we were to do it on our own what if they want you to take and use random number generator to simulate rolling two number of cubes well you'd only have to roll it one time for the first number cube roll the second time for the second number cube boom you're done easy math now we're going to let each trial represent two rolls to the number cube so we're going to press roll the cubes and what's going to be the relative frequency of rolling sum of 7 in your results let's take look here we're gonna roll it once and actually they said that they had two different sets so they rolled two cubes ten times the relative frequency of rolling to seven well it never happened it was an impossible event and we could have known that with our theory i'm sorry sum over seven my bad it wants sum of seven and sum of seven that is going to end up being we've got one right here this seven and then we have two and looks like two out of the ten times you ended up getting sum of seven and rolling your number cubes let's keep going as we learn how to simulate simple events zoom in some for you here simulation is defined as an experiment that is designed to model one or more events so you're not actually doing the event you're simulating the event the simulations often model events that would be too difficult or too time consuming or impractical in real life suppose cereal company places prize in one out of every three boxes of cereal you can design simulation that models whether or not box of cereal you buy is going to contain price the event consists of randomly selecting cereal box to simulate the event you can design an experiment that has the same probability of excess in this case the probability of excess is going to be one out of three because one out of every three boxes is going to contain prize let's keep going one way you could design this simulation is to design spinner that has probability of excesses an outcome is one out of three so the blue would be your winner the reds would be your failures and then when you move through that you would spin and see how many times it happened another way you could design simulation is to use number cube because number cube has six sides you could rewrite the problem of that as probability of one-third as the equivalent fraction with the denominator six so in this simulation you could say define successes by rolling two of the six faces so maybe you said that one or two represents cereal box with prize everything else you're gonna lose and what do we have here so what is gonna be not getting prize three four five and six you're not going to get prize so we use six number cube to simulate three option outcome where one and two are winners and four five and six are going to be losers check our answer and the checks now we're going to take and drag the icon to represent the related event to the model that can be used to correctly simulate the event and we've got let's zoom down some spinner with four equal sections or one toss of coin so here your favorite book picking your favorite book out of four books randomly aside that's going to be that your facebook baseball team has three out of four chance of winning once again that's going to be out of four section rain 50 chance of rain that's toss of coin there heads or tails half chance of the girls basketball team wins their first game heads or tails and certain carbon marble is randomly chosen from bag containing four marbles there you have your four outcomes all those are going to be correct but what if we had to simulate compound events as with simple events you can design simulation to simulate compound event coins number cubes and spinners are often used to simulate events design and simulation you need to do each of the following define what outcome represents and determine its successor failure rate and then define what each trial represents watch this animation and it's going to explain that in more detail for you you are asked to use simulation to estimate the probability that all three cubs in litter of three tiger cubs will be females begin by designing the simulation for each cub there are two possible outcomes female or male so choose tool with two possible outcomes such as tossing coin there are three cubs in the litter so use three coins in each trial let heads represent female cub and tails represent male cub next perform the simulation and record your data the table shows the results of 100 trials of the simulation now use the data to find the experimental probability the experimental probability is expressed as the ratio of the number of experimental successes to the number of experimental attempts in this case you want to estimate the probability of three female cubs there were 14 experimental successes in the simulation and the number of experimental attempts is equal to the number of trials finally simplify the ratio based on the simulation the estimated probability that all three tiger cubs will be female is seven over fifty or fourteen percent well that looks pretty darn simple really think you all can handle doing that let's find out continuing with that learned activity it says there's many ways to simulate what situation that involves probability and it says select the topics and represent the to learn about simulations that for probability events related to it so if we wanted to look at the weather 50 chance of rain you could use coin toss where you got heads and tails you could use spinner divide it into an even number of sections where half of them say it's going to rain half of them say it's not going to rain and then we can take and we got to go back home but doing marbles suppose you have bag with equal number of red and blue marbles and green marbles select assimilation to determine the probability of randomly selecting marble well we could take the number cube and we could say that two of them represent one caller two of them represent the next caller and then two of them represent the caller after that or we could take and do spinner and say that one of divided into three equal sections and say that each one of those sections equals one of the colors going back to that number cube want to make sure you all understand so one and two would mean you'd get red maybe three and four could represent blue and four and five that could represent green so if you score this is maybe four that means that four would be the mark color of marble that would have been picked on that trip and then in football here suppose that on average professional football player makes two out of three of his field goals once again that's me very similar to the last one we did we could take and say that the question is what is the probability he makes two goals in row row well that would mean that he would have two out of six it can be four out of six chance he has one out of three chance that's two out of three chance that means he has four out of six chance of getting two in row so we'll select each trial that consists of that you could also use spinner where you have two sections that are good one sections that are missed and then spin that two times to find out the odds that way example number one wants us to simulate compound event local grocery store sells cereal in two packs for special price the probability box containing price is one out of three design and simulate an event that represents the probability of randomly selecting 2-pack that contains box prize in both boxes and then it says run that simulation 10 times what's the simulated probability of getting the prize in both boxes of cereal so let's see what they do they ended up taking spinner where you got two red and one blue section and this one you've got one in three chance of getting prize out of box of cereal so they took the one blue section is going to be your winner the two red sections those are going to represent failure otherwise known as not getting prize from here we're going to keep moving one trial consists of spinning the spinner twice so let's move through the slides and see how we're going to do it so we've got packages box one and box two and whether or not you're going to get two prizes so when we took and spun it the first time we spun we spun once and we got prize we spun the second time we did not get prize we continued doing it we spun once we got pro we did not get prize we spun second time we did get prize again we spun once no prize we spent second time no prize and you continue through this process until you get to having your whole table filled out here and you can see that what are the odds of getting prize twice there was one two times out of ten times this is going to be your experimental probability two out of ten also known as one and five chance so move back over here how many times did you have prize in both boxes twice so the estimated probability of selecting two pack that contains prize in both boxes out of ten is going to be two out of ten also known as 0.2 also known as 20 that's all there is to do with these now it's gonna be your turn work extra example one on your own where there's three and four chance of it snowing to having enough to have snow day on thursday and three out of four chance of snow day on friday student wants to run simulation to estimate the probability that there will be snow day on both days how can student model situation of snow day on both days option each trial consists of tossing coin once they're twice no because we're dealing with three out of four chance so that's not going to happen because the coin is one out of two chance and then says each trial consists of spinning spinner with four equal sizes twice label three of them with snow and label the remaining with no snow success rate of both landing on snow represents winning and failure represents one or both days landing on no snow that one right there that is going to end up being your answer check it and it's winner continue moving down we're going to take ins the table shows results of 10 trials of the compound event and represents day where there is snow day and represents day where there is no snow day according to the simulation what is the experimental probability of having snow days on both thursday and friday we had one two three four days out of ten where they ended up getting snow day on both four over ten is the same thing as saying for two out of five as fraction so there's my fraction four out of ten is also the same as 4 10 or 0.4 which is the same thing as 40 chance of getting snow on both days what did miss here one two three four all right my answer was right the textbook is wrong on this they're saying it's three out of ten when clearly trial one trial six trial seven and trial ten all gave you two snow days on this one the textbook is wrong my answer was right for example number two we have computer simulation that was designed to simulate rolling number cube multiple times until all the positive even numbers were rolled the relative frequency bar graph shows the number of roles needed for the computer to roll all of the even numbers what is the simulated probability that eight or fewer roles were needed to obtain all the even numbers of the number cube first we need to find the sum of the relative frequencies that indicate either six seven or eight rolls was needed in order to obtain all of the even numbers so they said the for their formula the probability of the event the mint being less than or equal to eight rolls is gonna be the probability of getting into six rolls plus the probability of getting in seven rolls plus the probability of getting in eight rolls so scrolling down here we can see next that the probability of six was going to be 0.8 you can see that occurred here at 0.8 the probability of 7 was going to be 2.0 and you could follow it up it lines up with the frequency of i'm sorry 0.20 and the probability of 8 was it 0.10 on the frequency chart right there it's maybe 0.10 from there all you have to do is add those up together which gives you 0.4 chance of scoring all your even numbers in the 10 rolls and that is also ended up as number of your roles and that also is going to end up being the same as saying 40 chance the relative frequency has the same value as the experimental probability so the probability is that it takes eight or fewer roles in order to obtain all of your even numbers on the number cube is going to be at 40 percent now you get one more problem to work on this will be the last problem of this lesson for extra example number two emily designs computer simulation with 50 trials and uses data to create graph the graph shows the relative frequency of the number of times coin was tossed in order to land four times it heads four times in row use graph what is the probability the coin will need to be tossed seven eight or nine times in order to land heads four times write your answer as percent go ahead and stop the video and work that now so i've set up my two formulas the probability of seven or eight or nine is equal to the probability of seven plus the probability of an 8 plus the probability of 9. so you can see the probability of 7 that ended up being at 0.14 plus the probability of an 8 right here ended up at being 0.16 plus the probability of 9 right here ended up being 0.10 when we add all those together we're going to get let's see 14 plus 16 is going to be 30 plus 10 is going to be 40 so it's going to be 0.40 and that of course is going to be the same thing as saying 40 percent messed up my circle there come on baby delete that circle let's try that again there's my 40 percent jump back over to the textbook put in our 40 and check and the answer checks that's gonna be it for this lesson get busy working on your homework and let me know if you have any questions in class tomorrow