Math 033More Probability Models

Understanding probability models is essential for making sense of randomness, predicting outcomes, and drawing conclusions from data—skills that are foundational in statistics, science, and real-world decision-making. In this course, students will build on concepts like random sampling and variance to explore more advanced probability models, enabling them to interpret patterns, evaluate uncertainty, and apply mathematical reasoning to real-life situations.

Experimental Probability versus Theoretical Probability

This video explains the differences between experimental and theoretical probablility.

Transcript

0:00 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, one, two, three, four, five, six, seven, I'm not gonna draw all of them but you get the general idea.

0:14 There are going to be 50 magenta marbles, and there's also going to be 50 blue marbles. And what you do is, you have these 100 marbles in there, half of them magenta, half of them blue.

0:36 And before picking a marble out, and you're gonna be blindfolded when you pick a marble out, you shake the bag really good to, so you think, mix them up a little bit. And so, if you were to say theoretically, what is the probability, if you stuck your hand in and you're not looking, what is the probability of picking a magenta?

1:00 I feel the need to write magenta in magenta. What is the probability of picking a magenta marble? Well, theoretically there's 100 equally likely possibilities, there's 100 marbles in the bag. And 50 of them involve picking a magenta.

1:18 So, 50 out of 100, when 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.

1:32 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 the marbles, and then you see what marble color you picked, and you put it back in, and then you do it again.

1:48 And so, let's say that after every time you put your hand in the bag and you take something out of the bag, and you observe what it is, we're gonna call that an experiment. So, after 10 experiments, let's say that you have picked out seven magenta and three blue.

2:23 So, is this strange that out of the first 10 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.

2:38 If the true probability of picking out a magenta is 1/2, it's definitely possible that you could still pick out seven magenta, that just happened to be what your fingers touched. And this isn't a lot of experiments, it's completely reasonable that out of 10,

2:51 Yeah, you could have, later on in statistics we'll define these things in more detail, but there's enough variation in where you might pick that you're not going to always get, especially with only 10 experiments, you're not definitely going to get exactly 1/2.

3:09 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 question what I did here when I calculated this theoretical probability. But let's say you have a lot of time on your hands.

3:27 And let's say after 10,000 trials here, after 10,000 experiments, and remember the 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.

3:48 And so, let's say after 10,000 experiments, you get 7,000 magenta. Actually, I'm gonna do slightly different numbers, so let me make it even more extreme. Let's say you get 8,000 magenta and you have 2,000 blue.

4:13 Now this is interesting, because here 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 is completely reasonable that, hey you know, I got seven magenta and three blue instead of five and five, but now you've done 10,000.

4:33 You would've expected if this was the true probability, you would've expected that half of these would've been magenta, only 5,000 magenta and 5,000 blue, but you got 8,000 magenta.

4:43 Now, this is within the realm of possibility if the true probability of picking a magenta is 1/2, but it's very unlikely that you would've gotten this result with this many experiments, this many trials if the true probability was 1/2.

4:57 Here your experimental probability is showing, look, out of 10,000 trials, experimental probability here is you had 10,000 trials, or 10,000 experiments I guess you could say. And in 8,000 of them, you got a magenta marble.

5:26 And so, this is going to be 80%, or 8/10. So, there seems to be a difference here. 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 one half, t's very low likelihood that you would've gotten this many magenta.

5:49 So, when you think about it you're like, "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.

6:01 Well, why would this happen, I mixed up the bag every time. And there're some different possibilities; maybe the blue marbles are slightly heavier, and so when you shake the bag up enough, the blue marbles settled to the bottom, and you're more likely to pick a magenta marble.

6:17 Maybe the blue marbles have a slightly different texture to them, in which case, maybe they slip out of your hands, or they're less likely to be gripped on, and so you're more likely to pick a magenta.

6:29 So, I don't know the explanation, I don't know what's going on in that bag, but if I thought theoretically that the probability should be 1/2, because half of the marbles are magenta, but I'm seeing through my experiments that 80% of what I'm picking out,

6:43 especially if I did 10,000 of them, if I did this 10,000 times, well, 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, a magenta versus a blue. Something else must be going on.

Attribution

Khan Academy. (2015, February 17). Comparing Theoretical to Experimental Probabilities| 7th grade | Khan Academy. https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-probability-statistics/cc-7th-theoretical-and-experimental-probability/v/comparing-theoretical-to-experimental-probabilites

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