Discrete probability distributions

Welcome to Discrete Probability Distributions!

Hi there! Welcome to one of the most practical chapters in your A Level Statistics journey. In this section, we are going to learn how to model and predict the "chance" of things happening when we are dealing with things we can count. Whether it's the number of heads in ten coin flips or the number of faulty lightbulbs in a box, discrete probability distributions give us the tools to handle it all.

Don't worry if the name sounds a bit intimidating—we’re just going to break it down into simple, logical steps. By the end of these notes, you'll be able to spot a Binomial Distribution from a mile away and use your calculator like a pro to find the answers!

1. Understanding Discrete Random Variables

Before we look at the distributions, we need to understand the main character: the Discrete Random Variable.

Random Variable (X): A variable whose value is determined by the outcome of a random event. We usually use a capital letter \( X \) to represent the variable itself, and a lowercase \( x \) to represent a specific value it could take.

Discrete: This means the variable can only take specific, separate values (like 0, 1, 2, 3). You can’t have 2.5 siblings or flip a coin 4.7 times! If you can count it on your fingers, it’s probably discrete.

Probability Distribution Tables

A probability distribution is just a complete list of all possible values of \( X \) and their matching probabilities. It’s most commonly shown as a table.

Example: Let \( X \) be the score on a fair four-sided die.
\( x \): 1, 2, 3, 4
\( P(X=x) \): 0.25, 0.25, 0.25, 0.25

The Golden Rule

For any discrete probability distribution, the sum of all probabilities must equal 1.
Mathematically, this is written as: \( \sum P(X=x) = 1 \).
If your probabilities don't add up to 1, something has gone wrong!

Probability Functions

Sometimes, instead of a table, you'll be given a formula (a probability function) to calculate the chance of each value.
Example: \( P(X=x) = kx \) for \( x = 1, 2, 3 \).
To find \( k \), you would solve: \( k(1) + k(2) + k(3) = 1 \), so \( 6k = 1 \) and \( k = 1/6 \).

Quick Review Box:
• Discrete = Countable values (no decimals).
• Random Variable = The outcome of a random "experiment".
• Sum of Probabilities = Always exactly 1.

Key Takeaway: A discrete probability distribution is just a map that tells you every possible outcome and how likely it is to happen.

2. The Binomial Distribution

This is the "star of the show" for this chapter. The Binomial Distribution is a specific type of distribution used when we repeat a task several times and count how many "successes" we get.

When can we use the Binomial Model? (The B.I.N.S. Test)

To use the Binomial distribution, the situation must pass the B.I.N.S. test. If one of these fails, you can't use it!

B - Binary: There are only two possible outcomes for each trial (usually called Success and Failure).
I - Independent: The outcome of one trial does not affect the next one (like flipping a coin).
N - Number: There is a fixed number of trials (\( n \)).
S - Success: The probability of success (\( p \)) must stay the same for every trial.

Did you know? The name "Binomial" comes from "bi" (meaning two) because each trial only has two results—like a bicycle has two wheels!

Notation

We write \( X \sim B(n, p) \) to show that \( X \) follows a Binomial distribution with \( n \) trials and probability of success \( p \).

Key Takeaway: Use B.I.N.S. to check if a situation is Binomial. If the probability changes (like picking socks from a drawer without putting them back), it is not Binomial.

3. Calculating Binomial Probabilities

There are two ways to find probabilities: using the formula or using your calculator. For the OCR syllabus, you need to be comfortable with both!

The Formula

To find the probability of getting exactly \( x \) successes in \( n \) trials:
\( P(X=x) = \binom{n}{x} p^x (1-p)^{n-x} \)

• \( \binom{n}{x} \): This is the "nCr" button on your calculator. It tells you how many ways you can choose \( x \) items from \( n \).
• \( p^x \): The probability of success to the power of how many successes you want.
• \( (1-p)^{n-x} \): The probability of failure to the power of the remaining trials.

Using Your Calculator

Most modern calculators (like the Casio ClassWiz) have specific modes for this:
1. Binomial PD (Probability Density): Use this for "exactly" questions, e.g., \( P(X = 3) \).
2. Binomial CD (Cumulative Distribution): Use this for "range" questions, e.g., \( P(X \le 3) \).

Watch Your Inequalities!

This is the most common place where students lose marks. Because the data is discrete, the "equal to" part matters a lot!
• \( P(X \le 5) \): Includes 0, 1, 2, 3, 4, and 5. (Use Bcd directly).
• \( P(X < 5) \): This is the same as \( P(X \le 4) \).
• \( P(X \ge 5) \): This is \( 1 - P(X \le 4) \).
• \( P(X > 5) \): This is \( 1 - P(X \le 5) \).

Analogy: If a "Strictly under 18" club exists, an 18-year-old cannot get in. In the same way, \( X < 5 \) does not include the number 5.

Common Mistake to Avoid: Don't forget that \( (1-p) \) is just the probability of failure. Sometimes people call it \( q \). So, \( p + q = 1 \).

Key Takeaway: Always draw a small number line (0, 1, 2, 3, 4, 5...) and circle the numbers you want. This helps you figure out which cumulative probability to subtract from 1.

4. Mean and Variance of a Binomial Distribution

Even though you don't need to calculate the mean and variance for general discrete distributions in this chapter, you do need them for the Binomial Distribution!

The Mean (\( \mu \))

The mean is the "average" number of successes you would expect if you ran the experiment many times.
\( \mu = np \)

Example: If you flip a fair coin 100 times, you expect \( 100 \times 0.5 = 50 \) heads. Simple!

The Variance (\( \sigma^2 \))

The variance tells you how much the results are spread out.
\( \sigma^2 = np(1-p) \)
(Or \( npq \), where \( q \) is the probability of failure).

Quick Review:
• Mean (\( \mu \)) = \( np \)
• Variance (\( \sigma^2 \)) = \( npq \)
• Standard Deviation (\( \sigma \)) = \( \sqrt{npq} \)

Key Takeaway: These formulas are very helpful for describing the distribution and are essential when you move on to Normal Approximations later in the course.

Summary Checklist

Before you finish this chapter, make sure you can:
1. Explain what a Discrete Random Variable is.
2. Check that the sum of probabilities in a table or function equals 1.
3. Use the B.I.N.S. criteria to identify a Binomial Distribution.
4. Calculate exact Binomial probabilities using the formula.
5. Use your calculator to find cumulative probabilities (like \( P(X \le x) \)).
6. Calculate the Mean and Variance of a Binomial Distribution using \( np \) and \( npq \).

Don't worry if this seems tricky at first! The more you practice identifying \( n \) and \( p \) in word problems, the more natural it will become. You've got this!