Unit 6 Module 9 6 Simulate Chance Events 7 th Grade

so we're getting to the last lesson in module nine and this one lesson 9.6 is to simulate chance events and learning Target can design simulation to represent simple or compound event and use the results of simulation to find the experimental probability the vocabulary that we're going to be learning is simulation okay and what is simulation simulation is an experiment that is designed to model one or more events and simulations often model events that can be difficult time consuming or impractical to perform in real life okay suppose Serial company you know it says that there is prize in one out of three boxes that you are buying okay so suppose it's like it's placing prize in one out of every three of it serial boxes you can design simulation that models whether or not box of cereal you buy will contain prize okay so we're going to look at this and am going to model using spinner and I'm going to model using number cube okay the the event consists of randomly selecting Serial box to to simulate the event you can design an experiment that has the same probability of success all right so it has the same probability of success and in this case the probability of success is one out of three because one out of the three boxes contains prize okay so one way you can design simulation is to design spinner that has the probability of successful outcome as one3 so in this case the spinner will have three equal size sections so I've created this each of them it is one third right and so there is equal probability of Landing in any of these when it is spun so therefore I'm creating this it will have three equal size sections and in simulation success is defined as the spinner landing on blue because we said one out of three so I've just made the spinner so if it lands on Blue there's going to be success all right because that is one out of the three and that represents selecting box with prize this one the blue is representing box with the prize because the prize is in one out of every three box boxes all right and failure is defined as the spinner landing on red and represents selecting box without prize so two out of three is the probability of failure right because the prize is only in one out of three boxes so can represent by the spinner like this and can simulate that event all right another way that can do this is you know can design simulation is to use number cube okay number cubes are normally they are like one through six and because the number cube has six sides I'm going to rewrite the probability which we said one out of the three boxes is going to have prize right so one out of three boxes is going to have price so can design simulation by writing an equivalent fraction with denominator of six because know that the number cube has six sides so can easily do this can rewrite this as six so this would represent my six all right and know that 3 * 2 is six so 1 * 2 is going to give me two out of six faces will represent selecting box with prize okay so in this simulation success is defined by Rolling two of the six faces and represents selecting box with prize and failure is defined by Rolling four of the six faces represent six faces and represents selecting box without price you can determine which two faces are successes suppose decide that in number cube my roles you know if it falls on one or if it falls on two that is going to be My Success okay or that is going to be the serial box with price so what roles are going to then so if I'm saying one and two over here because we know that number cube has six sides so if I'm saying that one and two if roll one or two because two out of two out of six is going to be my 1/3 and know that one of the three boxes has the prize so if say one and two are representing the two and that's the success of the prize then what is the not success you know what not going to be successful of or what is the failure of finding the priz and then that would be three four five six so four out of six would be where do not find the price so this is how can simulate that event okay so am now going to show you this table and I'm just going to give you an example okay here are some examples and based on this example we are going to see couple of things all right so we are going to simulate each event by placing an in that column so there are five different events that we can see okay 1 2 3 four five different events that we can see in this table and we going to choose the model that can be used to correctly simulate each event by placing an in that column okay so let's look at this we've got an event we are trying to simulate it and which of these methods can use so we've got spinner with four equal size sections we have four equal size sections okay and then we have coin toss and know for coint TOS you can only have two outcomes heads or tails all right so let's look at these events and think about which one can use for simulation in this context so the first one your favorite book out of your four books being randomly assigned for book report so when we're talking about four different books obviously we are going to need four equal size sections so then therefore this one would be the correct Choice okay so this one would be the correct Choice spinner with four equal size sections let's look at the second one your favorite baseball team has three out of four probability of winning so we need four different sections so we are definitely going to choose the spinner with four equal size sections in this case all right let's look at the next one we have one half chance one one out of two ratio chance girl soccer team wins its first game so since there are two options the coin toss would be great because in coin toss you have the probability either head tail so there are only two choices so because of that in this one would use this for simulation all right forecast shows 50% chance of rain so 50% 50 out of 100 which is 1 half so 50% chance of rain 50% no chance of rain right so therefore we just need two options over here so for this the coin toss would be the best way to simulate that event right and then let's look at the last one marble is randomly chosen from bag contain containing four different color marbles now again we need four different sections because we're talking about four different colored marbles so because of that would choose the spinner with four equal size sections so just an example of how can create simulation and spinner or coin toss or number cube these are all great ways to create you know to to create simulation or to simulate chance events okay so let's quickly discuss I'm going to just take you to the board over here so that you can see the question so that we can design simulation all right so you know as with simple events you can design simulation to simulate compound event and as said coins number cubes and Spinners are often used to simulate events to design simulation you need to do each of the following okay so you need to Define what each outcome represents and determine if it is success or failure and then Define what each trial represents so let's look at this question over here we have to design simulation all right so suppose each tiger cub born in litter of cubs has an equal chance of being female or male okay suppose each Tiger cup born in lit of cubs has an equal chance of being female or male and in litter of three tiger cubs estimate the probability that all three Cubs will be females okay so let's look at this the step one do have to design simulation and for each cup there are two possible outcomes you can have female or you can have male right so since it's just two options what would be the best way to design simulation is to toss coin because coin has two possible outcomes okay and because there are three Cubs in the litter each trial represents tossing three coins you can choose to let heads represent female cup and Tails represent male cup okay so heads again female or we can have Tails which is representing male cup so in order to perform the simulation suppose have table which shows the results of 100 trials after the simulation so let me just go ahead show you the table and let's say this is what get okay so this table is showing the results of 100 trials of the simulation so these are the outcomes and this is relative frequency three females and we got zero males 14 two females one male 33 one female two males 41 zero females three males 12 so that's relative frequency that we got after 100 trials of that tossing the coin okay so if add up all this this is 100 so now I'm going to see the what was the question the question was asking that you know what what what will happen if if we can estimate that all three Cubs will be females so let's look at the relative frequency of three Cubs being female and let's see if the experimental probability which has the same ratio as the relative frequency okay so once we did that what did we see we got 14 right because three females so for the simulation data for the 100 trials of three females zero males my frequency is 14 and so 14 which is out of 100 okay so 14 out of 100 and so then what am going to do I'm going to simplify it and when simplify that ratio divide by two and get 7 over 50 right so if simplify that ratio get 7 over 50 and then based on this simulation what is the estimated probability that all three tiger cubs will be female 7 over 50 what is that that's nothing but 14% okay so the answer would be 1400 it would be 7 over 50 or it would be so here we go 1400 7 out of 50 or 14% So based on the simulation the estimated probability that all three tiger cubs will be female is what we just calculated that is 14% okay now am going to give you some scenarios and you know I'm going to just put the question up here so it's easy for you to see and refer back to if need be but I'm going to give you couple of scenarios and then we can decide how to simulate that event okay so let's look at this one again there are many ways to simulate compound events some examples I'm going to show you now okay suppose suppose that during the spring time it rains 50% of the days what is the chance that it will rain two days in row this spring okay so two days in row so I'm just going to show you by pulling up what can do because it's talking about 50% that is 50% raining 50% not raining so best thing we can either use coin toss okay or or we can also use spinner so if if you look at this okay try to get this out of the way okay all right so let's look at this one so Coos let heads represent rain let Tails represent no rain because we know that the possibility of rain or no rain is just two different you know outcomes that are there so because of that coin toss would be great okay so each trial consists of two dosses of coin because we're talking about rain in two days okay so two tosses of coin and successful event is represented by tossing two heads because we said let heads represent rain okay and then whereas the spinner is concerned can have four sections with equal sections of rain and no rain so each as you can see is 25% it's you know four different equal paths two with rain two with no rain and each trial consists of two spins and successful trial is the pointer landing on rain two times in row so this is great simulation of weather all right let's do another example and here's one and let me try to there we go all right so here is one where we have marbles that we talking about and suppose you have bag of marble with an equal number of red blue and green marbles so now that means we have three different colors okay what is the probability of randomly selecting red mark from the bag three times in row with replacement with replacement means the marble you know once it's picked up it is put back in the bag if it was without replacement if remove it it is out and then whatever I'm left with from that I'm selecting but it says with replacement that means am putting it back into the bag after picking that marble okay so there are two good ways to do this because they are three different sections we can use the number cube all right and so because number cube basically what does it have it has out of six parts so what we can do is we can have two unique numbers over there to represent red and each trial is going to consist of rolling the number cube three times and success is landing on the two specified numbers okay the two specified numbers three times in row okay so three times in row let me try to pull this up so you can see this little better there we go all right so let's say chose one and two on the number cube to represent you know red so if it if it lands on one and two it's success if it lands on 3 four 5 six it is not successful so that that's basically what this is okay and then we can use spinner as you can see have green red blue three different sections successful trial is the pointer landing on red three times in row okay so because we talked about three times in row with replacement for both of these landing on the on the two specified numbers three times in row and same thing over here three times in row so this is great way to show simulation all right now let's talk about another example so that you understand how we are creating the simulations let's talk about football just do this okay so suppose that on average professional football kicker Mak makes two out of three of his field goals from the 40 yard line what is the probability that he makes two field goals in row okay so when look at this one we are going to assign four so we're going to use number cube and spinner again in this case because it's talking about 2/3 so in number cube can definitely use that you know my equivalent ratio in number que have out of six so can do that over three okay and we know that two of them are going to be now failure in this case and four are going to be successful in this case because we are talking about two out of three of his field goals from the 40 yard line so we're going to assign four of the numbers on the number cube to represent the success rate of two out of three because two out of three is What four out of six all right right so let's say the first four numbers in the number cube are representing the success and each trial consists of rolling the cube twice and success is represented by landing on any of those assigned numbers two times in row so as said one two three four on the number cube let's say the first four are representing success so if it lands on any of those four numbers two times in row then that is success all right if it lands on five or six it'll be not success successful okay so it depends on which numbers you are designating designating to represent the two out of the three on the number cube the spinner so again we can you know have three parts and each trial consists of two spins and successful trial is the poter landing on good two times in row because two out of three all right so this is how we simulate that now let us go ahead and quickly do an example on simulating compound events all right so again let's say have question and my question is local grocery store sells cereal in two packs for special price the probability of box containing price is 1/3 okay so probability of this of containing prize is one3 okay design and simulate an event that estimates the probability of randomly selecting two pack that contains prize in both boxes and then run the simulation 10 times okay so so we said 1/3 was the probability of it containing prize so first thing I'm going to do is I'm going to design simulation and this is compound event because the success of the first event represents one box in the two pack containing prize and the success of the second event represents the other box in the two pack containing price so one way to design simulation is to use spinner to represent each event and because the probability of each box containing prize is one3 we are just going to design spinner like this okay so have done you know have designed this spinner with three equal size sections you can let one section such as blue this is going the blue is going to represent the success of selecting box that contains price so blue is going to be the success because one out of three okay so blue is the success and the other two SE sections represent the failure of selecting box that does not contain price so both of these over here because two out of three means it's failure because it has to be one out of three so that would be the success and the other two would be the failure okay so one trial consists of spinning the spinner twice so I'm just going to show sample result in table okay so if draw out table and we are we said that we are going to spin it like 10 times okay so we have our pack okay and box one box two box one box two and then we have both prizes both prizes okay and we had talked about spinning it how many times 10 times right so if it if the one trial comes spinning the spinner twice and let's just you know spin it 10 times because that's what the question was we're going to run the simulation 10 times so that is why here are my packs one 2 three 4 5 5 6 7 8 9 10 and if just make table like this and then conduct my experiment okay and so let's say this is box one is this box two no all right and here we have our simulations and I'm just giving an example and this is correct and no and six and let's do seven and 8 no and nine and then have 10 okay and so let's look at both prizes which ones have both prizes that have been selected so here no no no yes no yes no no no and no so let's look at that table and these are the sample results which have been shown in the table once again again my question was local grocery store sells cereal in two packs for special price the probability of box containing priz is oneir design and simulate an event that estimates the probability of randomly selecting two pack that contains prize in both boxes run the simulation 10 times and what is the simulated probability of getting prize in both boxes of of Serial okay so what we're going to do here we know that the simulation was how much we ran it 10 times and how much was it that we got yes so I'm going to circle the ones that have yes this one and this one so out of 10 how many are yes it is 210 so this is going to be two out of 10 okay this is 210th right this is also written as 210th and can change this to percent okay so would go you know 210th is equal to what over 100 this will be * 10 * 10 that is 20 so 20 over 100 so this is also 20% it's also 20% so the estimated probability of selecting two pack containing prize in both boxes based on the 10 simulated trials is 210th or 210th as decimal or 20% all right so that's how we do the calculation so let's go ahead and do one more example and in this one we are going to do an interpretation of you know interpret the simulations of compound events okay so let's say give you problem which is up here okay let's say give you problem like this all right so computer simulation was designed to simulate rolling number cube multiple times until all of the possible 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 so here is my okay so there is my graph okay so what is the simulated probability that eight or fewer roles are needed to obtain all of the even numbers on number cube so if I'm looking at this graph need to look at eight or less okay so we have eight or less so what we're going to do now is we are going to find the sum of the relative frequencies that indicate that six seven or eight rows were needed to obtain all of the even numbers okay so let me show you how you would do this calculation so eight or less so we are going to be looking at you know probability okay less than equal to 8 right and we're talking about the roles is equal to what so it's going to be the probability of six 7even and eight because eight or less right so probability of six plus probability of 7 plus probability of 8 okay and as can see over here what are the three probabilities okay so this one six is Right six is going to be my 800 seven is 210 all right and 8 is 1200 okay so look at these three and then all do is am going to add them all so have the probability of six was how much 800 + 210 + 1200 okay and then when add all this up am going to get 400 and 400 is nothing but 40% so this is 40% and that is my answer so the relative frequency ratio has the same value as the experimental probability so the simulated probability that it takes eight or fewer roles to obtain all of the even numbers on number cube is 40% all right so once again this is all about your simulating chance events okay or simulations often model events that can be difficult timec consuming or impr impractical as said to perform in the real life because of the size of that so that's when we create these simulations and and that this would conclude our model module nine for the you know all the unit on probability right so have fabulous day everyone