Binomial Distribution: Summary

“Green…green…yellow…green…green…”

For what felt like the thousandth time, Gregor Mendel was looking at peas in the monastery garden, and noting down whether each one was yellow…or green

He was just finishing another experiment, in which he observed the colour of a pea, and then repeated that trial hundreds of times. 

Ultimately, he wanted to know the probability of a pea being green.

And so, with his tallies for the most recent experiment complete, he took his total frequencies and calculated an experimental probability…or, in other words, a relative frequency…

The relative frequency from this most recent experiment was 0.75, to 2 decimal places.

Now, every time Mendel ran this experiment, he found pretty much the same probability…about 0.75, or a 75% chance of a pea being green.

And this discovery had surprisingly enormous implications…

Because Mendel wasn’t just looking at peas for fun…

Instead, he had a new theory about how all living things inherit traits from their parents…

Whether that’s dogs inheriting white spots…

Humans inheriting red hair…

Or peas inheriting their colour.

And Mendel’s result, that his peas had a 75% chance of being green, heavily implied that Mendel’s groundbreaking theory was right.

As a result, scientists eventually accepted his theory. 

In fact, Mendel’s theory of inheritance is still the strongest explanation for inheritance we have. And it’s so widely accepted that it’s even taught in schools!

However, Mendel’s famous pea experiments have a dark, dark secret…

In 1936, about 70 years after Mendel’s experiments, a famous British statistician called Ronald Fisher accused Mendel of faking his results!

And actually, scientists and statisticians still argue about whether Mendel faked his results. 

But, to understand Fisher’s fiery accusation, we need to spend some time learning about a new type of probability distribution, called a binomial distribution…

To understand Ronald Fisher’s bold claim about Mendel’s work, we need to learn about a special type of probability distribution, called a binomial distribution.

But before we can do that, we first need to learn what a binomial trial is.

So, a binomial trial is just any trial in which there are only two outcomes.

For example, flipping a coin once is a binomial trial…

Because there are exactly two outcomes…heads or tails.

Seeing whether one pea is green or yellow is also a binomial trial…

Because there are exactly two outcomes…green or yellow.

And spinning this spinner once is also a binomial trial… [⅔ red, ⅓ blue]

Because there are exactly two outcomes…red or blue.

So now, which of these are binomial trials?

Each of these are binomial trials, since they have exactly two outcomes.

Now, there are many more examples of binomial trials as well.

For example, if we roll a die and focus on whether we get a 5 or not…

Then that is a binomial trial, since there are now only two outcomes…

Either we do get a 5, or we don’t!

Equally, if we roll a die 3 times and focus on whether we get at least two 5s…

Then that is still a binomial trial, since there are still only two outcomes…

Either we do get two or more 5s, or we don’t!

In fact, in any experiment…

We can create a binomial trial by just focusing on whether something does happen…

Or doesn’t happen!

So, which of these are binomial trials?

Each of these are binomial trials, since they have exactly two outcomes.

Finally, out there in the big wide world of more advanced mathematics…

Binomial trials are more commonly known as Bernoulli trials [pronounced ‘bu-newey’].

And that’s because they’re named after the 17th-century Swiss mathematician Jacob Bernoulli [pronounced ‘yakob’], who first wrote about these trials in a very famous book on probability, called Ars Conjectandi. 

But, for the rest of our course, we’re going to keep referring to them as binomial trials.

And that’s because they’re closely related to binomial distributions…

Which we’ll look at…next!

To sum up, a binomial trial is…

A binomial trial is any trial which has exactly two outcomes. 

And we can create binomial trials in any experiment by just focusing on whether something does happen….or does not happen.