What is Binomial Distribution?

Under the harsh light of his desk lamp, Ronald Fisher studied Gregor Mendel’s results. 

Fisher sensed something was amiss with the values Mendel published.

But he didn’t have the time or patience to plant and then categorise thousands of peas. 

Fortunately, since Mendel’s experiments were built from binomial trials…

Where we focus on whether a pea is green…or not…

Fisher knew there was a special way he could interrogate Mendel’s experiments…without planting a single pea…

So, to start off, if the probability of a pea being green is 0.75…

And we look at 80 peas…

Then how many green peas do you think we’re most likely to see: 50, 55, or 60?

If the probability of a pea being green is 0.75, then we’d expect to find about 60 green peas.

But now, is it possible that we could find 55 green peas instead?

Yes, it’s perfectly possible that we’d just happen to only find 55 green peas!

What about 50 green peas? Is it possible we’d find only 50?

It’s a little less likely but, yeah, it’s still totally possible that we’d only find 50 green peas. 

Alright, let’s decrease that number way more…right down to 5…a measly 5 green peas.

If the probability of a pea being green is 0.75, and we look at 80 green peas…

Is it possible that we’d only find 5 green peas?

This time, it’s really unlikely that we’d only find 5 green peas…

But it is still possible!

In fact, it’s possible that we’d find any number of green peas…right the way from 0…to 80.

And this brings us back to Fisher, and how he was able to explore Mendel’s experiment without having to harvest a single pea from a monastery garden… 

Fisher knew how to calculate the theoretical probability that Mendel would find each number of green peas in his experiments

For example, he could calculate the probability of finding 60 green peas…

He could calculate the probability of finding 65 green peas…

And he could calculate the probability of finding 53 green peas.

So in addition to these probabilities, what other probabilities could he calculate?

Fisher could calculate the probability of finding 50 green peas…

The probability of finding 67 green peas…

The probability of finding 80 green peas…

And even the probability of finding 0 green peas

In fact, Fisher could calculate the probability of finding any number of green peas between 0 and the full 80

Except, the probabilities for some of these outcomes are so low that we can’t even see the bars!

So, to remind us they’re still there, from this point on we’re going to put a little dot at the top of each of our bars, like this…

And with that, Fisher was able to construct a complete probability distribution…

Which, in this case, we’ve represented as a diagram!

Armed with this distribution, Fisher was able see how many green peas Mendel was likely to find in each of his experiments…

And he could do that from the comfort of his dingy office…without having to plant a single pea!

For example, which outcome has a probability of about 0.1?

Finding 61 green peas out of 80 has a probability of about 0.1.

Now, since we started with a binomial trial…observing whether a pea is green or yellow…

We say that this probability distribution…is a binomial probability distribution.

Or, just a binomial distribution for short!

And we’ll explore other binomial distributions, and the features they all share, next…

But to sum up for now, in Mendel’s experiment, you could observe any number of green peas.

And so, it’s possible to calculate the probability of finding each number.

Then, if you put all of those probabilities in one place…like a diagram…

You get a special type of probability distribution, called…

You get a special type of probability distribution called a binomial distribution.