_{Up Learn – A Level maths (edexcel) – Binomial Distributions}

_{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 Trials (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)}

^{When Can We Use Binomial Distributions? (free trial)}

^{Independent Trials (free trial)}

## Probability Distributions

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2. The Things We Record

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3. Probability Tables

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4. Probability Bar Graphs

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5. Probability Distribution

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6. Uniform Distribution

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7. Different Experiment, Same Sample Space

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8. Probability Distribution Notation

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9. Probability Lists

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10. Probability Distribution Notation: The Outcomes

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11. Probability Distribution Notation: P(X = x)

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12. Piecewise Notation

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13. Probability Functions

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14. Completing the Probability Function

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15. Non-Numerical Outcomes

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16. Random Variables

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2. Binomial Trials

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3. Binomial Distributions

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4. Features of Binomial Distributions

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5. Setting Up a Binomial Expression

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6. Finding Probabilities in a Binomial Distribution

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7. Our Old Friend the Calculator

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8. Bringing in the Random Variable

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9. When Can We Use Binomial Distributions?

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10. Independent Trials

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11. The Magic of Binomial Distributions

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12. Binomial Distribution Notation

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13. Changing the Probability of Success

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14. Changing the Number of Trials

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15. The Mendel-Fisher Controversy

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2. The Manual Approach

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3. Cumulative Distributions

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4. Plotting Cumulative Distributions

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5. Other Cumulative Distributions

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6. Finding Cumulative Probabilities With Your Calculator

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7. Finding P(X<x)

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8. Finding P(X>x)

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9. Finding P(X≥x)

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10. Working Backwards

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11. Gombaud’s Game

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2. Continuous Random Variables

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3. Discrete Does Not Mean Finite

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4. A Problem With Continuous Random Variables

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5. Ranges of Outcomes

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6. Finding Probabilities With Histograms

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7. Probability Histograms

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8. Supercharging Histograms With Curves

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9. Representing Probability With Height vs Area

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2. A Normal Distribution

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3. Mean and Standard Deviation

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4. Varying the Standard Deviation

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5. The Asymptote of the Normal Distribution

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6. Normal Distribution Probability Zones – Part 1

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7. Equal To or Not Equal To

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8. Normal Distribution Probability Zones – Part 2

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9. Calculating Probabilities – Part 1

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10. Calculating Probabilities – Part 2

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11. The Four Question Types

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12. Relating areas under a normal curve

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13. Inverse Problems

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14. Normal Distribution Notation

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15. Variance and the Standard Deviation

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16. The Standard Normal Distribution

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17. Finding the Mean

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18. Finding the Standard Deviation

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19. Using Phi

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2. When Binomial Distributions Behave Normally

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3. Limitations of Normal Approximations

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4. The Continuity Correction

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5. Finding the Mean of the Normal Approximation

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6. Finding the Variance of the Normal Approximation

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“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.

WHAT YOU GET

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**Exclusive Practice Papers**

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