Distributions

The Casio CG50 and its newer version, the CG100, make light work of working out distributions. The calculator provides a clear representation of what the distribution the student is investigating leads to, fostering a better understanding of the statistics.
 

Understanding Binomial and Normal Distributions — and How to Use Your Calculator to Solve Probability Problems

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Statistics is everywhere — from predicting election outcomes to understanding how likely it is that your delivery arrives late. Two of the most important tools in statistics are the binomial distribution and the normal distribution. Each tells us about different types of data, and in this blog post, we’ll explore what they are, when to use them, and how to solve probability problems with a calculator.

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🎯 What’s the Difference Between Binomial and Normal Distributions?

Let’s start with the basics.

🔢 Binomial Distribution

The binomial distribution describes the probability of a fixed number of successes in a set number of independent trials, each with the same probability of success.

Requirements:

• Fixed number of trials, n

• Only two outcomes: success or failure

• Constant probability of success, p

• Independent trials

Example:

You flip a fair coin 10 times. What’s the probability of getting exactly 6 heads?

This is a binomial problem:

• $n = 10$,

• $p = 0.5$,

• $X = \text{number of heads} \sim B(10, 0.5)$

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📈 Normal Distribution

The normal distribution describes continuous data that clusters around a mean. It’s that famous bell-shaped curve — symmetric, with most values close to the average.

Requirements:

• Continuous data

• Symmetrical, bell-shaped distribution

• Defined by mean $\mu$ and standard deviation $\sigma$

Example:

The height of adult men is normally distributed with a mean of 175 cm and a standard deviation of 8 cm. What’s the probability a randomly selected man is taller than 183 cm?

This is a normal distribution problem:

• $X \sim N(175, 8^2)$

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🧮 Using a Calculator to Solve Probabilities

Most modern scientific and graphical calculators (like the Casio fx-991EX or the TI-84) can calculate binomial and normal probabilities easily.

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📊 Solving a Binomial Problem on a Calculator

Example:
A basketball player scores a penalty shot 80% of the time. What is the probability she scores exactly 7 out of 10 shots?

• $X \sim B(10, 0.8)$

On a Casio fx-991EX:

1. Mode → Statistics

2. Choose Binomial CD (for cumulative) or Binomial PD (for individual probability)

3. Enter:

   • x = 7

   • n = 10

   • p = 0.8

Result:
$P(X = 7) = 0.2013$ (or similar, depending on calculator model)

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Cumulative Example:
What’s the probability she scores at most 7 goals?

Use Binomial CD, with $x = 7$:
$P(X \leq 7)$

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🔔 Solving a Normal Distribution Problem

Example:
Heights are normally distributed: $X \sim N(175, 8^2)$. What’s the probability someone is taller than 183 cm?

1. Use Normal CD

2. Set:

   • Lower = 183

   • Upper = 9999 (to simulate ∞)

   • $\mu = 175$, $\sigma = 8$

Result:
$P(X > 183) ≈ 0.1587$

This means about 15.9% of people are taller than 183 cm.

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🔄 Linking the Two: When Binomial Approximates Normal

If you have a binomial problem with large $n$, you can often approximate it with a normal distribution using:

• $\mu = np$

• $\sigma = \sqrt{np(1-p)}$

Don’t forget the continuity correction!

So:

• $P(X = 7) \to P(6.5 < X < 7.5)$

• $P(X \leq 7) \to P(X < 7.5)$

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🧠 Summary Table

Distribution	Type	Example	Calculator Mode
Binomial	Discrete	Coin flips, success/fail outcomes	Binomial PD / CD
Normal	Continuous	Heights, weights, measurement errors	Normal CD
Normal Approx.	Approx. Binomial (large n)	Many trials, moderate p	Use mean/SD with correction

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🧪 Final Thoughts

Understanding and using binomial and normal distributions is a key part of statistics, especially for GCSE and A-Level Maths and Sciences. Once you grasp when and how to apply them — and learn to harness the power of your calculator — you can confidently tackle a wide range of real-world and exam problems.