Discrete probability distributions

Welcome to Discrete Probability Distributions!

In this chapter, we are moving from basic probability into the world of Statistical Distributions. This is where we stop looking at just one-off events and start looking at the "big picture" of what outcomes are likely to happen over many trials. Think of it like moving from looking at a single raindrop to understanding the whole rainstorm! This is a vital skill for careers in science, finance, and even game design.

1. What is a Discrete Random Variable?

Before we dive into the math, let's break down that scary-sounding name: Discrete Random Variable.

Variable: A value that can change (usually called \(X\)).
Random: We don’t know the exact outcome until it happens.
Discrete: The values are separate and distinct. You can count them (like 0, 1, 2, 3). You can’t have "2.5 children" or "3.14 heads" in a coin toss.

Real-World Analogy:
Imagine you are shooting 5 basketball free throws. The number of baskets you make is a Discrete Random Variable. You could make 0, 1, 2, 3, 4, or 5 baskets. You can't make 2.7 baskets!

Important Point: We use a capital \(X\) for the name of the variable (e.g., "Number of baskets") and a lowercase \(x\) for the specific value (e.g., "making 3 baskets"). So, \(P(X = x)\) just means "The probability that the number of baskets is equal to a specific number."

2. Describing the Distribution

There are two main ways the OCR syllabus expects you to see these distributions: as a table or as a formula.

The Probability Table

This is the most common way to see a distribution. It lists every possible outcome and the chance of it happening.

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

The Golden Rule: The sum of all probabilities in a distribution must equal 1.
\(\sum P(X = x) = 1\)
If you add up the bottom row of your table and don't get 1.0, something is wrong!

The Probability Function (Formula)

Sometimes, the probability is given as a little machine. You plug in \(x\), and it spits out the probability.
Example: \(P(X = x) = kx\) for \(x = 1, 2, 3\).

To find the value of \(k\), you would plug in the values:
When \(x=1\), \(P = 1k\)
When \(x=2\), \(P = 2k\)
When \(x=3\), \(P = 3k\)
Since they must add to 1: \(1k + 2k + 3k = 1\), so \(6k = 1\), which means \(k = 1/6\).

Key Takeaway: Whether it's a table or a formula, the total probability is always 1. Use this fact to find missing values!

3. The Binomial Distribution

The Binomial Distribution is a special type of discrete distribution that pops up everywhere. It’s used when you have a set number of "trials" and each trial has only two outcomes: Success or Failure.

When can we use the Binomial Model?

Don’t worry if this seems tricky; just remember the mnemonic B.I.N.S.! To use the Binomial distribution, the scenario must meet these four conditions:

B - Binary: Only two possible outcomes (Success or Failure).
I - Independent: One trial doesn't affect the next (like flipping a coin).
N - Number of trials: There is a fixed number of trials (\(n\)).
S - Success probability: The probability of success (\(p\)) stays the same every time.

Notation: We write this as \(X \sim B(n, p)\).
This is math shorthand for: "The variable \(X\) follows a Binomial distribution with \(n\) trials and a probability of success \(p\)."

Did you know? Sampling "without replacement" (like picking names out of a hat and not putting them back) technically violates the Independence rule. However, if the population is very large, we often assume it's Binomial anyway because the change in probability is tiny!

4. Calculating Binomial Probabilities

You need to be able to find the probability of getting exactly \(x\) successes using two methods: the formula and your calculator.

The Formula

\(P(X = x) = \binom{n}{x} \times p^x \times (1 - p)^{n-x}\)

Let's break this down into "human language":
\(\binom{n}{x}\): This is the number of different ways the successes could happen (found using the \(nCr\) button on your calculator).
\(p^x\): The probability of success raised to the number of successes you want.
\((1 - p)^{n-x}\): The probability of failure raised to the number of failures you'll have left over.

Example: If you flip a biased coin 5 times (\(n=5\)) where the chance of heads is 0.6 (\(p=0.6\)), what is the chance of getting exactly 3 heads?
\(P(X = 3) = \binom{5}{3} \times 0.6^3 \times 0.4^2\)
\(P(X = 3) = 10 \times 0.216 \times 0.16 = 0.3456\)

Using Your Calculator (The Pro Way)

In your exam, speed is key! Most modern scientific calculators have a Binomial PD (Probability Density) mode for "exactly" questions and Binomial CD (Cumulative Distribution) for "less than or equal to" questions.

Common Mistake to Avoid: Calculators usually calculate \(P(X \le x)\). If the question asks for \(P(X > 3)\), you must calculate \(1 - P(X \le 3)\). Always check the direction of the inequality!

Quick Review Box:
\(P(X = x)\) Use Binomial PD
\(P(X \le x)\) Use Binomial CD
\(P(X < x)\) Change it to \(P(X \le x-1)\) then use Binomial CD

5. Summary and Key Takeaways

1. Discrete Random Variables represent countable outcomes where the total probability is always 1.
2. B.I.N.S. is your checklist to see if a situation is Binomial (Binary, Independent, fixed Number, same Success prob).
3. \(X \sim B(n, p)\) is the notation you must use in your working out.
4. Mean and Variance: For the H230 AS Level, you do not need to calculate the mean or variance of these distributions. Focus on finding probabilities!

Final Tip: When explaining "assumptions" in context, don't just say "it's independent." Say "We assume that one person catching the flu is independent of the next person catching it." Context is everything!