So to start, we just want to introduce this idea of simple probability.
And at its core, the idea here is that the probability of what we'll call for now event A and event
A could be anything we choose, but the probability of event A occurring is equivalent to the number
of matching outcomes divided by the number of total outcomes.
Now, of course, we need to be specific about what we mean by matching outcomes and total outcomes,
but the idea is that matching outcomes are any outcomes that satisfy event A and total outcomes are
all of the possible equally likely outcomes that could occur in the scenario.
So for instance, let's take a very simple example of flipping a coin, and we're looking for the probability
that we get heads when we flip the coin.
So in this very simple scenario, we flip a two sided coin one time.
On one side is heads, on the other side is tails.
And we want to know the probability that we get heads when we flip that coin.
So in this case, event a from our formula is the event that we get heads.
So we could either say event A is heads and we could calculate the probability of event A like this,
or we could also right it this way, the probability of getting.
Heads.
We don't have to write event A specifically, we can just write the probability of getting heads.
So the probability of flipping heads is equal to well, what are the total outcomes in this scenario?
We can either flip heads or we can flip tails.
There's two possibilities two equally likely outcomes from flipping this coin.
So the total number of all possible outcomes in the scenario is two heads or tails.
This idea then of matching outcomes is the number of outcomes that match our scenario here of Event
A, which is heads.
We're interested in flipping heads.
And so only one of the two scenarios matches that event.
If we flip tails, it doesn't match this event of getting heads only getting heads when we flip the
coin matches this event.
So the number of matching outcomes is simply one, which means that the probability of flipping heads
when we flip a coin one time is one half or.
50%, one half is equal to 50%.
There's a 50% chance we get heads when we flip a coin.
What about a different example here where we roll a six sided die one time and we want a two?
What's the probability that we get a two when we roll that die?
We could write this scenario as the probability of getting a two.
So we might write it like this.
The probability of rolling a two is going to be equal to if we're rolling a six sided die one time.
Of course, there are six total possible outcomes, all of which are equally likely.
We could roll a one, a two, three or four of five or six.
There are six different outcomes we could get when we roll that die.
So six equally likely outcomes, six total outcomes.
And the number of those outcomes that match our criteria here of rolling a two is of course just one,
because if we roll a one, a three for five or six, that doesn't match our criteria.
Only rolling a two matches our criteria.
So there's only one possible outcome, one matching outcome to the one that we're looking for.
So the probability of rolling a two is one over six.
We can also do something a little more complicated where we say, what is the probability of rolling
an even number if we roll that six sided die?
Well, we might write that as the probability of rolling.
An even number.
And we already know from this previous example that there are six possible outcomes when we roll that
die one, two, three, four, five and six.
And of those possible outcomes, only the numbers two, four and six are even numbers one, three and
five are odd numbers.
So the only matching outcomes are the outcomes where we roll a to a four or a six, that's three possibilities.
So the probability in this case is three matching outcomes over six total outcomes.
And then whenever we can, we always want to simplify our fraction, three over six is equivalent to
one half.
So we could write this as one half or.
50%.
So there's a 50% chance there's a one in two chance that we roll an even number.
And then just one more example where we roll a die.
What's the probability that we roll a number on that die that is greater than two?
So we could write this a few ways, but let's just say the probability that X is greater than two,
meaning our die roll results in a number that is greater than two.
Well, we know there are six possible outcomes one, two, three, four, five or six when we roll that
die.
But what about the matching outcomes?
Well, the numbers on the die that are greater than two are the numbers three, four, five and six.
The numbers one and two are not greater than two.
Only three, four, five and six are values greater than two, which means there are four matching outcomes.
And so our probability is four and six or two in.
Three.
A two thirds chance, a two and three chance, or an approximately 67% chance of rolling a number greater
than two When we roll that six sided die one time and we're looking for a number greater than two.
So we understand this general idea that probability of event A, the probability of flipping heads or
rolling A to rolling an even number, rolling a number greater than two.
It's just the likelihood that any of these events happens, given the scenario that we're setting up.
So just think about probability as likelihood of that event occurring.
We also want to be aware at this point, though, of this idea of complimentary probability or the complementary
event and the complement of any event is the scenario where that event does not occur.
And we would write it like this If we're looking for the complement of a we might write it as P.
Of a prime and this would indicate the complement.
So here, if we're looking for the complimentary event of flipping heads, we could write it this way.
Heads and then prime.
This means the complement of the heads event or the probability that we don't flip heads when we flip
a coin one time.
Well, there's two ways to calculate this.
We could, of course, think about going back to our original probability formula and calculating it
manually or from scratch.
So the probability that we don't flip heads.
Well, if we flip a coin one time, we know there are two possible outcomes that stays the same.
The possible outcomes are heads or tails.
The only outcome out of heads or tails that meets this event, not heads, is tails.
And so there's one way that we can match this event, not heads, and that's by flipping tails.
And so the probability of knot heads is one half.
And that makes sense because knot heads, of course, is equal to just tails if we're thinking about
a coin.
And so we could really also write this as the probability of flipping tails and that's one half.
Or to take another example here, we're saying the probability of rolling a two when we roll a six sided
die one time is one over six.
So the probability of.
Not rolling it to is going to be, again, six total outcomes because we're rolling a die one time and
we know that there are six possibilities there, but then not getting a two.
Well, the scenarios that match that are rolling a one, three, four, five or six five different possible
matching outcomes.
So this complementary event here of not to the probability of that complimentary event is five over
six and we could go ahead and calculate that for these other two examples here.
But the thing that we want to notice is that in both of our examples so far, the probability of the
original event and its complement, when we add them together, we always get one or 100%.
So the probability of heads was one half.
The probability of not heads was one half.
So when we add those two things together, we get one half plus one half and we get one.
Or we could call that 100%.
In this scenario, we had the event rolling it two and then the complementary event not rolling it two.
If we add the probabilities of the two complementary events together, we get one over six plus five
over six is equal to six over six or just one or.
100%.
And it turns out that this is always going to be the case.
The sum of the probability of an event and the probability of its complement will always be a total
of one or 100%.
And that makes sense, really, because an event and its complement represents the entire universe of
possibilities, right?
When we roll a die.
There's six possible outcomes if we're talking about the probability of rolling a two and then not rolling
a two.
That, of course, is going to represent all the possibilities when we're flipping a coin, the probability
of getting heads or not getting heads, in other words, getting anything other than heads is going
to represent the entire universe of possibilities.
Over here, when we roll a die and we're looking for an even number, the probability of getting an
even number and then any possibility other than an even number is, of course, going to add to 100%.
It's like saying the probability of this one section of possible outcomes plus the probability of all
the other possible outcomes.
That, of course, is going to give us all possible outcomes or 100% of possible outcomes, which is
why we always end up with these 100% figures when we add the probability of the event and its complement,
which of course leads us into this idea that probability itself is always a value between zero.
And one where zero, a probability of zero means that the event will not occur or that there's a 0%
chance that that event will occur all the way up to one which says that the event will definitely occur,
that there's 100% chance that the event will occur.
And so then we have everything in between.
If we think about this as a percentage where this is 0% all the way over to 100%, it almost makes it
easier to imagine a 0% chance that an event occurs.
A 100% chance that an event occurs.
And then everything in between, like on the coin flip, where there's a 50% chance that we get heads,
a 50% chance we get tails that's right in the middle of this probability scale.
So what this leads us to understand then, is that because summing the probability of an event and its
complement is always going to lead us to this 100% figure, we know that we can always find the probability
of the complement as one minus the probability of the event itself.
Or we could write that in reverse as the probability of event A is equal to one minus the probability
of the compliment.
These formulas are equivalent.
They're just reworked a little bit or rewritten in different ways.
In other words, if we know that the probability of rolling a two is one over six and we want to calculate
the probability of not rolling a two, all we have to do is say the probability of not rolling a two
has to be equal to one minus the probability of rolling a two.
So one minus the probability of rolling a two is one over six.
So we get one minus one over six or.
Five over six, five over six has to be the probability of not rolling it too, which is in fact exactly
what we saw here.
Now that we understand this idea of the probability of an event and the probability of its complement,
and the fact that those two probabilities will always sum to 100%, let's circle back to this original
probability formula and talk about this idea of experimental versus expected probability.
So we just talked about the idea of flipping a coin with two sides, heads and tails, and we said that
the probability of flipping heads was going to be equal to a one in two chance one matching outcome
over two total outcomes or a 50% chance of flipping heads on any one coin flip.
This 50% figure that we calculated and this formula here for the probability of any event A is specifically
what we call expected probability, or we also call it theoretical probability.
It's the probability we expect to have under the given scenario.
But think about this problem here.
Let's say that we're flipping a coin and we flip it five times.
So we flip the coin once and we get heads.
We flip the coin a second time, we get heads, we flip the coin, a third time, we get heads again,
we flip the coin of fourth time, we get heads and then we flip the coin a fifth time and it comes up
tails this time.
So out of our five flips, we got heads, heads, heads, heads, tails.
If we look at this little experiment that we ran, what it would imply to us is that if we flip a coin
five times every five times we flip the coin, we're going to get heads four out of those five times.
And so we might look at this experiment and say that there is a four in five chance or in percentage,
an 80% chance and 80% probability that we get heads when we flip a coin.
Now, of course, intuitively we know that's not true, but if we run this experiment, it makes it
look like there's an 80% chance that we get heads when we flip a coin.
So what exactly is going on here?
Well, we call this 80% number here.
We call this experimental probability.
And experimental probability can vary widely because if we run an experiment where we continue to flip
a coin over and over and over and over again, every time we flip the coin, the probability is going
to change.
In other words, we can see the change happen in real time, right?
So if we flipped the coin five times and we got four out of five outcomes as heads, we would say that
it looks like there's an 80% chance that we get heads when we flip the coin.
Well, if on our very next flip, we flip the coin a sixth time and we get heads, there's now a five
out of six chance of five out of six chance that we get heads when we flip a coin.
But if on our sixth flip we get tails, there's now a four out of six chance that we get heads as we're
flipping this coin, this is an approximately 83% chance of heads.
And this is an approximately.
67% chance of heads, which makes sense because if our probability was sitting at 80% after we flipped
a coin five times and got four of them to come up heads, if we flip heads again, then our experimental
probability that we get heads is increasing.
It now looks like we have an 83% chance of getting heads.
If we flip tails, then that probability of getting heads is going to go down.
It drops from 80% chance to 67% chance.
So we can see our experimental probability moving and changing every time we redo this experiment.
So every single time we flip the coin as we continue this experiment, our experimental probability
of getting heads is going to move in one direction or another.
So that experimental probability can vary widely, but the expected probability stays fixed.
The expected probability is given by this formula here of matching outcomes divided by total outcomes.
And it's the kind of probability that we're used to thinking about What this law of large numbers tells
us is that as we run our experiment more and more and more and more times, our experimental probability
is going to get closer and closer and closer to our expected probability.
So to illustrate that, we could sketch out a graph where we have here along the horizontal axis, our
number of coin flips, and here along the vertical axis, the percentage of flips where we flip heads.
If we think about taking this horizontal axis out to more and more and more experiments.
So here we've only flipped the coin ten times.
Here we've flipped it 50 times all the way up to this far side where we've now flipped the coin 200
times.
What we see is that our data should approach this 50% expected probability because of course we know
the probability of getting heads on any particular coin flip should be 50% as long as we have a fair
coin.
So we should have this line here.
But if we were to sketch in some data from our coin flipping experiment, it's possible it might look
something like this.
When we have a very small number of flips, our percentage of heads might be far from this 50% expected
probability figure and in general, with each additional flip.
So maybe this is where we are at ten flips at the 11th flip, we might move further from this 50% line
or closer to it, and we might go farther away from the line for a while.
But the concept is that eventually, as long as I flip the coin enough times, this data will revert
to the mean.
It'll pull closer and closer into this line.
And even though it might wiggle around it, like we see the data here wiggling around this line, eventually
it's going to converge to this 50% line, which is our expected probability.
So this is our expected.
Probability.
And this is our experimental probability.
Which wiggles in both directions up and down, around expected probability, until eventually it converges
to be closer and closer and closer to expected probability and experimental probability will get closer
to expected probability the further we go out along this horizontal axis.
In other words, the more times we flip the coin.
That is the law of large numbers.
It tells us that experimental probability will get closer and closer to expected probability.
As we run the experiment more and more.
The reason we bring this up experimental versus expected probability is to illustrate basically two
main points.
The first point being that the fewer times we run the experiment, the more variability we can expect
and the further away we can be from expected probability without even necessarily realizing it.
And so this is a good lesson for us to learn when we are running experiments and calculating probability,
the idea being that the smaller the sample size we have, the fewer number of times we flip the coin,
the fewer number of times we roll the die.
If our data is such that we can't necessarily calculate this perfect expected probability number, we
can't fill in this formula.
And instead we have to experiment to see what our probability figure is looking like.
Then we need to realize that the chance of getting an inaccurate figure for probability is higher.
When we're taking smaller samples, when we're sampling a smaller number of data points, which means
our conclusion then is that in order to get a more accurate estimate of probability, we want to take
the largest sample that we possibly can.
If we're flipping a coin, we want to flip it as many times as we possibly can to get a better estimate
of actual expected probability instead of flipping it just a small number of times and relying on whatever
our experimental probability figure happens to be.
So again, our our two main points being that we expect highly variable experimental probability when
we have small samples or small data sets.
And as we increase the size of that data set, the law of large numbers tells us that we'll get closer
to expected probability.
And therefore our conclusion is that we always want to use the largest data set that we possibly can
or that's reasonable to use so that we can get the clearest possible picture of expected probability.