Distributions: Binomial and Normal

Introduction - Distributions: Binomial and Normal

As you study A Level Maths, especially in Statistics, you will come across the term “distributions.” A distribution is simply a way of showing how often different values occur in a set of data.

It allows us to see patterns in data. Such as are the values around a certain value or are they spread out or even if there are any unusual values.

It is like a snapshot of your data which tells the story of how the values are laid out.

Distributions relate to how the values in a population are distributed. Two of the important distributions in mathematics are the normal distribution and the binomial distribution. Understanding these distributions will help you solve many questions in probability.

The binomial distribution relates to a fixed number of events and that the probability of success does not change. The normal distribution refers to continuous data and gives a “bell curve” shape.

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Review statistics and probability principles before moving on to distributions.

Binomial Distribution

The binomial distribution applies to trials of a fixed experiment. There are two results possible in each trial: success and failure. Some examples of this are throwing a coin, testing whether a light bulb is good, or passing a test.

In order to use the binomial distribution, there are three factors necessary:

n = the number of trials

p = the probability of success on a single trial

x = the number of successes

The probability formula is:

P(X = k) \;=\; \binom{n}{k}\,p^{\,k}\,(1 – p)^{\,n – k}

Example of Binomial Distribution

You flip a coin 5 times. What is the probability of getting exactly 3 heads?

Here we have a binomial distribution.

The number of trials is $n = 5$
The number of successes is $k = 3$
The probability of success is $p = 0.5$

Apply the formula:

P(X=3) \;=\; \binom{5}{3} \Bigl(\tfrac12\Bigr)^3 \Bigl(\tfrac12\Bigr)^{5-3} \;=\; \binom{5}{3} \Bigl(\tfrac12\Bigr)^5.

The final answer is $\displaystyle \frac{5}{16}$

Normal Distribution

We use the normal distribution when working with continuous data which is generally symmetrical about the mean. When plotted the data forms a bell shaped curve and most of the values are around the mean. There are some which occur at the ends but these are extreme values.

Key features:

Mean (μ): The average value.
Standard deviation (σ): How spread out the data is.
68–95–99.7 Rule: About 68% of values lie within one standard deviation of the mean, 95% within two, and 99.7% within three.

The normal distribution is used in exams to calculate probabilities, estimate outcomes, and understand variability.

Using the Normal Distribution in Exams

To calculate probabilities with the normal distribution, convert values to z-scores:

z = (x – μ) / σ

The z-score shows how many standard deviations a value is from the mean. Exam questions often ask for the probability of a value being above, below, or between certain points. You are most likely to use your calculator to then calculate these probabilities.

Using the Normal Distribution

To calculate probabilities with the normal distribution, change values to z-score values as follows:

$\boxed{\,z \;=\;\frac{x – \mu}{\sigma}\,}$

The z-score shows how many standard deviations a value is from the mean. Most questions on the exam ask for the probability of a value being above a certain value, or below a value, or between two points. In all cases, convert values to z-scores and then use tables or calculators to obtain individual probabilities.

Is there a Connection Between Binomial and Normal?

If the number of trials of a binomial distribution is large, and p is not too close to 0 or 1, the distribution can be approximated by a normal distribution. This is called the normal approximation. It greatly simplifies the calculations used in examinations.

Hints for Students

Decide whether the question is in terms of the binomial or the normal distribution.
Memorise the essential formulae and conversion of z-scores.
Draw diagrams for visualising probabilities which will assist in problem–solving.
Practice as many problems as possible to gain confidence in the procedure.
Ensure the normal approximation is appropriate when dealing with binomial problems.

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Conclusion

A knowledge of binomial and normal distributions will help you to understand the topics in probability and statistics. The binomial distribution develops from discrete events and the normal distribution applies to continuous data. If you understand its properties, formulas, and applications, you will improve your results in examinations. It will also be helpful to practise these techniques regularly and use visual devices such as probability trees or bell curves, in order to appreciate the techniques and increase confidence.

About the Author – S. Mahandru

S. Mahandru is Head of Maths at Exam.tips and has more than 15 years of experience in simplifying difficult subjects such as pure maths, mechanics and statistics. He gives worked examples, clear explanations and strategies to make students succeed.

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Finally, strengthen your understanding with conditional probability and counting techniques.

FAQS

What's the difference between binomial and normal distributions?

The binomial distribution is a distribution of discrete data from a fixed number of trials, each of which has two possible outcomes (success and failure). The other distribution mentioned, the normal distribution, is one of continuous data and has a “bell-shaped curve” shaped symmetrically around a mean. The binomial distribution is concerned with how many successes will occur, while the normal distribution is concerned with measuring the data and finding probabilities over a continuous range.

Under what conditions can a binomial distribution be approximated by a normal distribution?

The approximation of the binomial distribution by the normal distribution is possible only when the number of trials, n, becomes large, and the probability of success, p, is not near 0 or 1. This is the normal approximation and is useful in aiding the solution of probabilities on examinations, more especially on large populations.

How do I calculate probabilities in a normal distribution?

To calculate probabilities on a normal distribution, convert the x-value in the question to a z value using the following equation: z = (x – μ)/σ, where μ = mean and σ = standard deviation. The z value tells how many standard deviations from the mean the score is. You can then look up on tables or use your calculator to find the probability that this value is above or below or between certain points.