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## Randomness, probability, and simulation

# Experimental versus theoretical probability simulation

AP.STATS:

UNC‑2 (EU)

, UNC‑2.A (LO)

, UNC‑2.A.4 (EK)

, UNC‑2.A.5 (EK)

, UNC‑2.A.6 (EK)

, VAR‑4 (EU)

, VAR‑4.B (LO)

, VAR‑4.B.1 (EK)

CCSS.Math: , ## Video transcript

- [Instructor] 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 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 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 wanna show that on this graph here, we can plot it. And what this says is at a time, how many tosses do we wanna 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. And then as we flip, we're gonna 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's 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 in the second flip, it
looks like it was a tails. Because now the proportion 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 the fifth flip, it looks
like we got another heads and so now have three out
of five or 60% being heads. 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 gonna do another, well let's just do another 200
tosses and see what happens. So I'm just gonna keep tossing here and you can see, wow look
at this, there is 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 then 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 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
disproportionate number of tails. So I'm just gonna keep going, we're now at almost 800 tosses. And you see now we are converging. Now this is, we're gonna
cross 1,000 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 gonna 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 1,210, 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 gonna 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.