Experimental versus theoretical probability simulation | Probability | AP Statistics | Khan Academy

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 experiments, 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 McMillan 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 are 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 heads. We can make it unfair by changing this, but I'll stick with the 50% probability.

If 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 10 simulated flips of coins, with each one having a 50% chance of being heads. Then, as we flip, we're going to see our total proportion that are 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 your experimental probability after that one flip, you'd say, well, with only one experiment I got one heads, so it looks like 100% were heads. But then the second flip, it looks like it was a tails, because now the portion that was heads after two flips was 50%.

But then 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 at the fifth flip, it looks like we got another heads, and so now we have three out of five, or 60%, being heads.

And so the general takeaway here is when you have 1, 2, 3, 4, 5, or 6 experiments, it's completely plausible that your experimental proportion, your experimental probability, diverges from the real probability. This even continues all the way until we get to our ninth or 10th 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 of 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, and then a little run of heads, tails, and another run of heads.

And notice, even after 215 tosses, our experimental probability is still reasonably different from 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 gets 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 10 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 a disproportionate number of tails. So I'm just going to keep going. We're now at almost 800 tosses, and you see now we are converging. We now—this is—we're going to cross a thousand tosses soon, and you can see that our proportion here is now 51%. It's getting close!

Now we're at 50.6%, and I could just keep tossing. This is 1100, we're going to approach 1200 or 1300 flips right over here. But as you can see, as we get many, many, many more flips, it 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 to many, many more flips, now we're at 1210, we're getting pretty close to 50% of them turning out heads. But we could keep tossing it more and more and more, and 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.