Up Learn – A Level maths (edexcel) – Binomial Distributions
Binomial Distribution: Summary
Here’s a summary of everything you need to know about binomial distributions at A Level.
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More videos on Binomial Distributions:
Introduction to Binomial Distributions (free trial)
Binomial Distributions (free trial)
Features of Binomial Distributions (free trial)
Setting Up a Binomial Expression (free trial)
Finding Probabilities in a Binomial Distribution (free trial)
Our Old Friend the Calculator (free trial)
Bringing in the Random Variable (free trial)
Probability Distributions
2. The Things We Record (free trial)
3. Probability Tables (free trial)
4. Probability Bar Graphs (free trial)
5. Probability Distribution (free trial)
6. Uniform Distribution (free trial)
7. Different Experiment, Same Sample Space (free trial)
8. Probability Distribution Notation (free trial)
9. Probability Lists (free trial)
10. Probability Distribution Notation: The Outcomes (free trial)
11. Probability Distribution Notation: P(X = x) (free trial)
12. Piecewise Notation (free trial)
13. Probability Functions (free trial)
14. Completing the Probability Function (free trial)
15. Non-Numerical Outcomes (free trial)
16. Random Variables (free trial)
2. Binomial Trials (free trial)
3. Binomial Distributions (free trial)
4. Features of Binomial Distributions (free trial)
5. Setting Up a Binomial Expression (free trial)
6. Finding Probabilities in a Binomial Distribution (free trial)
7. Our Old Friend the Calculator (free trial)
8. Bringing in the Random Variable (free trial)
9. When Can We Use Binomial Distributions? (free trial)
10. Independent Trials (free trial)
11. The Magic of Binomial Distributions (free trial)
12. Binomial Distribution Notation (free trial)
13. Changing the Probability of Success (free trial)
14. Changing the Number of Trials (free trial)
15. The Mendel-Fisher Controversy (free trial)
2. The Manual Approach (free trial)
3. Cumulative Distributions (free trial)
4. Plotting Cumulative Distributions (free trial)
5. Other Cumulative Distributions (free trial)
6. Finding Cumulative Probabilities With Your Calculator (free trial)
7. Finding P(X<x) (free trial)
8. Finding P(X>x) (free trial)
9. Finding P(X≥x) (free trial)
10. Working Backwards (free trial)
11. Gombaud’s Game (free trial)
2. Continuous Random Variables (free trial)
3. Discrete Does Not Mean Finite (free trial)
4. A Problem With Continuous Random Variables (free trial)
5. Ranges of Outcomes (free trial)
6. Finding Probabilities With Histograms (free trial)
7. Probability Histograms (free trial)
8. Supercharging Histograms With Curves (free trial)
9. Representing Probability With Height vs Area (free trial)
2. A Normal Distribution (free trial)
3. Mean and Standard Deviation (free trial)
4. Varying the Standard Deviation (free trial)
5. The Asymptote of the Normal Distribution (free trial)
6. Normal Distribution Probability Zones – Part 1 (free trial)
7. Equal To or Not Equal To (free trial)
8. Normal Distribution Probability Zones – Part 2 (free trial)
9. Calculating Probabilities – Part 1 (free trial)
10. Calculating Probabilities – Part 2 (free trial)
11. The Four Question Types (free trial)
12. Relating areas under a normal curve (free trial)
13. Inverse Problems (free trial)
14. Normal Distribution Notation (free trial)
15. Variance and the Standard Deviation (free trial)
16. The Standard Normal Distribution (free trial)
17. Finding the Mean (free trial)
18. Finding the Standard Deviation (free trial)
19. Using Phi (free trial)
2. When Binomial Distributions Behave Normally (free trial)
3. Limitations of Normal Approximations (free trial)
4. The Continuity Correction (free trial)
5. Finding the Mean of the Normal Approximation (free trial)
6. Finding the Variance of the Normal Approximation (free trial)
“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.
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