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Theoretical and experimental probabilities


Video transcript

let's say that you've got a bag and in that bag you put a bunch of marbles so let's say you put 50 of these magenta marbles so 1 2 3 4 5 6 7 I'm not going to draw all of them but you get the general idea there are going to be 50 magenta marbles 15 magenta marbles and there's also going to be 50 blue marbles so 50 50 blue marbles and what you do is you have these 100 marbles in there half of the magenta half of them blue and before picking a marble out and you're going to be blindfolded when you pick a marble out you shake the bag you shake the bag really good too - so you think mix them up a little bit and so if you were to say well theoretically what is the probability if you stuck your hand in and you're not looking what is the probability of picking of picking a magenta I have to feel the need to write the word magenta in magenta what is the probability of picking a magenta marble well theoretically there's a hundred equally likely possibilities there's 100 marbles in the bag and 50 of them involve picking a magenta so 50 out of 100 and this is the same thing as a 1/2 probability so you could say well theoretically there is a 1/2 probability I just did the math if you say these are 100 equally likely possibilities 50 of them are picking magenta now let's say you actually start doing the experiment so you literally take a bag with 50 magenta marbles 50 blue marbles and then you start picking you start picking the marbles and then you see what marble color you picked and you put it back in and then you do it again and so let's say that after let's say that after what every time you you put your hand in the bag and you take something out of the out of the bag and you observe what it is what we're gonna call that an experiment so after after 10 experiments 10 experiments after 10 experiments let's say that you have picked out seven magenta seven magenta and three blue and three blue so does this cause is this is it strange that after ten it out of the first ten experiments you haven't picked out exactly half of them being magenta you've picked out seven magenta and then the other three were blue well no this is definitely a reasonable thing if the true probability of picking out of magenta is one-half it's definitely possible that you could still pick out seven magenta did that just happen to be what your fingers touched and this isn't a lot of experiments it's completely reasonable that out of ten yeah you could have a any other we later on in statistics we could all define these things in in more detail but there's enough variation in where you might pick that you're not going to always get especially with only ten experiments you're not definitely going to get exactly one half you can you you know instead of having five magenta it's completely reasonable to have seven magenta so this really wouldn't cause me a lot of pause I still wouldn't say hey I still wouldn't question what I did here when I calculated this theoretical probability but let's say and you let's say you have a lot of time on your hands and let's say after let's say after 10,000 trials here after 10,000 experiments and remember experiment you're sticking your hand in the bag without looking your fingers kind of feeling around picks out a marble and you observe the marble and you record what you found and so let's say after 10,000 experiments you get 7,000 7,000 magenta actually let me do slightly different numbers so let me make it even more extreme let's say you get 8,000 8,000 magenta and you have 2,000 blue 2,000 2,000 blue now this is interesting because here you're what you're what you're seeing experimentally seems to be very different and now you have a large number of trials right over here not just 10 10 it's completely reasonable it hey you know I get seven magenta and three balloons at a five and five now you've done ten thousand ten thousand it's definitely you know you would have expected if this was the true probability you would have expected that half of these would have been magenta only five thousand magenta and five thousand blue and you had eight thousand magenta now this is within the realm of possibility if the true probability of picking and magenta is 1/2 but it's not it's very unlikely that you would have gotten this result with this many experiments this many trials if the true probabilities 1/2 here your experimental probability your experimental probability is showing look out of out of 10,000 trials let's see let me write that here experimental probability experimental probability here is you had 10,000 trials and or 10,000 experiments I guess you could say and you and in 8,000 of them you got a magenta marble and so this is going to be 80% or 8/10 so 80% so there seems to be a difference here and the reason why I would take this more seriously is that you had a lot of trials here you did this 10,000 times if the true probability was 1/2 it's very low likelihood that you would have gotten this many magenta so when you think Michael okay well what did I what's going on here what are possible explanations for this this I wouldn't have fretted about after 10 experiments not a big deal but after 10,000 this would have caused me pause say well why would this happen I mixed up the bag every time and there are some different possibilities maybe the blue marbles are slightly heavier and so when you shake the bag up enough the blue marbles settle to the bottom and you're more likely to put pick a magenta marble maybe the blue marbles have a slightly different texture to them in which case maybe they slip out of your hands or there they're less likely to be gripped on and so you're more likely to pick a magenta so I don't know the explanation I don't know what's going on in that bag but if if I thought theoretically that the probability should be 1/2 because 1/2 the marbles are magenta but I'm seeing it through my experiments that 80% of what I'm picking out especially if I did 10,000 of if I did this 10,000 times well this is this is going to cause me some pause I would really start to think about whether it's truly equally likely for me to pick out a red magenta versus a blue something else must be going on