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CCSS.Math: , what we're going to do in this video is explore how experimental probability should get closer and closer to theoretical probability as we conduct more and more experiments or as we conduct more and more trials this is often referred to as the law of large numbers if we only have a few experiment it's very possible that our experimental probability could be different than our theoretical probability or even very different but as we have many many more experiments thousands millions billions of experiments the probability that the experimental and the theoretical probabilities are very different goes down dramatically but let's get an intuitive sense for it this right over here is a simulation created by Macmillan USA I'll provide the link as an annotation and what it does is it allows us to simulate many coin flips and figure out the proportion that our heads so right over here we can decide if we want our coin to be fair or not right now it says that we have a 50% probability of getting a heads we can make it unfair by changing this but I'll stick with the 50% probability we want to show that on this graph here we can plot it and what this says is at a time how many tosses do we want to take so let's say let's just start with 10 tosses so what this is going to do is take ten simulated flips of coins with each one having a 50% chance of being heads and then as we flip we're going to see our total proportion that our heads so let's just talk through this together so starting to toss and so what's going on here after 10 flips so as you see the first flip actually came out heads and if you wanted to say what's your experimental probability after that one flip you'd say well with only one experiment I got one head so it looks like 100% were heads but in the second flip it looks like it was a tails because now the proportion that was heads after two flips was 50% but in the third flip it looks like it was tails again because now only one out of three or 33% of the flips have resulted in heads now by the fourth flip we got a heads again getting us back to 50th percentile now the fifth lift it looks like we got another heads and so now we have three out of five or 60% being hit and so the general takeaway here is when you have one two three four five or six experiments it's completely plausible that your experimental proportion your experimental probability diverges from the real probability and this even continues all the way until we get to our ninth or tenth tosses but what happens if we do way more tosses so now I'm going to do another well let's just do another 200 tosses and see what happens so I'm just going to keep tossing here and you can see Wow look at this there was a big run getting a lot of heads right over here and then it looks like there's actually a run of getting a bunch of tails right over here then a little run of heads tails and another run of heads and notice even after 215 tosses our experimental probability has still is still reasonably different than our theoretical probability so let's do another 200 and see if we can converge these over time and what we're seeing in real time here should be the law of large numbers as our number of tosses get larger and larger and larger the probability that these two are very different goes down and down and down yes you will get moments where you could even get ten heads in a row or even 20 heads in a row but over time those will be balanced by the times where you're getting disproportionate number of tails so I'm just going to keep going we're not almost 800 tosses and you see now we are converging with now this is we're going to cross a thousand tosses soon and you can see that our proportion here is now fifty-one percent it's getting close now we're at fifty point six percent and I could just keep tossing this is 1,100 we're going to approach 1,200 or 1,300 flips right over here but as you can see as we get many many many more flips was actually valuable to see even after 200 flips that there was a difference in the proportion between what we got from the experiment and what you would theoretically expect but as we get too many men more flips now we're at 1210 we're getting pretty close to 50% of them turning out head we could keep tossing it more and more and more what we'll see is as we get larger and larger and larger it is likely that we're going to get closer and closer and closer to 50% it's not to say that it's impossible that we diverge again but the likelihood of diverging gets lower and lower and lower the more tosses the more experiments you make