A natural follow up to the Bernoulli distribution is what is called

the binomial distribution, and the binomial distribution happens

when you perform a set of Bernoulli trials.

So we start up with our basic Bernoulli, and we repeat it.

And we say how many successes did we get in those trials.

We often give the number of trials the letter little n.

Now, an example of a binomial would be to toss a coin ten times and

count the number of heads.

Each of the tosses is a Bernoulli experiment and

if we perform that Bernoulli experiment ten times and add up

the number of successes then we have what is called a binomial random variable.

And these binomials are commonly used, modeling distributions.

Now, there's one subtly here about the binomial random variable is that

the trials need to be what are termed independent events.

I want to briefly talk about what we mean by independence.

Independence is a basic modeling assumption in many probabilistic models.

And formally, what independence means is that the probability that an event A and

B happens can be written as the probability that

A happens times the probability that B happens.

And just going back to thinking of tossing a coin twice,

what's the probability I get two heads?

Well, given the coin tosses are independent, that's the probability

the first is a head times the probability that the second is a head.

And it's when you put in the word times there that you're using this independence

belief.

And so an independence assumption certainly makes

probability models more straightforward,

because we can essentially calculate probability though multiplication.

Another way of understanding independence is that knowing the one random variable

has occurred, that A has occurred, that occurrence provides no information about.

A subsequent random variable, which we'll call b.

So going back to the coin tossing, we're going to toss the coin twice.

If I tell you that I got a head the first time around how much

does knowing I got a head on the first coin toss tell me

about what's going to happen on the second coin toss?

The answer is absolute nothing because the events are independent.

So that's the nice way of thinking about whether two events are independent.

Does knowing the outcome of one impact the probability of a second event?

If it doesn't, then you can view the events as independent.

And that is often a simplifying assumption that has provided for

lots of probability models.

Now, it doesn't mean that it's true, though, and

it's something that the modeller would need to think about carefully,

maybe look at some data to see whether or not the independent's assumption was

realistic, but just note that in this binomial distribution,

we're taking these Bernoulli building blocks.

We're going to replicate, repeat that Bernoulli experiment, and

we're going to assume that the outcome from one to the other are independent.

So that's a binomial.

Number of successes in n independent Bernoulli trials.

So my example that I'm talking about here.

Is toss a fair coin ten times count the number of heads.

That would be, a binomial.

Going back to the drug development, if we had ten drugs under development, and

we believed that, whether or not they we're approved,

what independent event, and they all have the same probability

of being approved under all of those conditions.

Then the number of approvals that we're going to have could be modeled as

a binomial random variable.

But we'd need two things going on there,

one, independence of the decisions as to whether the drug was approved or not, and

two, it's the same probability for each of these Bernoulli events.

So whether or not that's the case, one would have to examine carefully.

But if it were, you'd be looking at a Bernoulli add up, but if it were,

you'd be looking at a binomial for the number of successes in those ten drugs.