Welcome to Statistical Distributions
Welcome to one of the most practical and scoring topics in CCEA AS Level Mathematics (Unit AS 2: Applied Mathematics). In this chapter, we explore how mathematics allows us to model real-world uncertainty, from predicting the number of defective items on a factory line to calculating your chances of rolling a six multiple times in a board game.
Don't worry if probability has felt confusing in the past. We will break everything down into bite-sized, step-by-step ideas. By the end of these notes, you will know exactly how to set up, calculate, and interpret discrete distributions with total confidence.
1. Discrete Random Variables
What is a Discrete Random Variable?
In statistics, an uppercase letter such as \(X\) represents a random variable—a quantity whose numerical outcome is determined by chance. We call it discrete because it can only take distinct, separate, countable values (typically whole numbers like \(0, 1, 2, 3\dots\)), with no values in between.
Analogy: Think of a staircase. You can stand on step 1, step 2, or step 3, but you cannot stand on step 2.47. That is discrete! (In contrast, continuous measurements like exact height or time are covered later in A2).
The Fundamental Rules (Axioms) of Probability
For any discrete random variable \(X\), two non-negotiable rules must always hold:
1. Bounded Probabilities: Every individual probability must be between 0 and 1 inclusive:
\(0 \le P(X = x) \le 1\)
2. Total Probability Equals 1: The sum of the probabilities of all possible mutually exclusive outcomes must equal exactly 1:
\(\sum P(X = x) = 1\)
Probability Distribution Tables
A discrete probability distribution can be written as a formula or laid out clearly in a table that lists each possible value \(x\) alongside its probability \(P(X = x)\).
Worked Example:
A discrete random variable \(X\) has the probability distribution given by \(P(X = 1) = 0.2\), \(P(X = 2) = 0.35\), \(P(X = 3) = k\), and \(P(X = 4) = 0.15\). Find the value of \(k\).
Solution:
Because all probabilities must sum to 1:
\(0.2 + 0.35 + k + 0.15 = 1\)
\(0.70 + k = 1\)
\(k = 0.30\)
The Discrete Uniform Distribution
A special case occurs when all possible outcomes are distinct and have the exact same likelihood of happening. This is called a discrete uniform distribution.
If a variable \(X\) has \(n\) equally likely outcomes, then the probability for each outcome is:
\(P(X = x) = \frac{1}{n}\)
Example: Rolling a single fair six-sided die. The possible values are \(x \in \{1, 2, 3, 4, 5, 6\}\), and because each face is equally likely, \(P(X = x) = \frac{1}{6}\) for every value.
Section 1 Key Takeaways:
• A discrete random variable takes distinct, separated values.
• The sum of all probabilities across the sample space is always \(\sum P(X = x) = 1\).
• If all \(n\) outcomes share an identical probability, it is a discrete uniform distribution where \(P(X = x) = \frac{1}{n}\).
2. The Binomial Distribution
What is the Binomial Distribution?
The Binomial Distribution is used when you carry out a fixed number of repeated trials, and each trial has only two possible outcomes (which we label as success or failure).
We write this using the notation:
\(X \sim B(n, p)\)
Where:
• \(X\) is the discrete random variable representing the total number of successes.
• \(n\) is the fixed number of trials (\(n \in \mathbb{N}\)).
• \(p\) is the constant probability of success on any single trial (\(0 < p < 1\)).
• \(q = 1 - p\) represents the probability of failure.
The 4 Essential Conditions: The "BINS" Memory Aid
In CCEA examinations, you may be asked to state the assumptions required for a binomial model. Always remember the mnemonic BINS:
• B — Binary outcomes: Each trial results in strictly two outcomes (Success or Failure).
• I — Independent trials: The outcome of one trial has no effect on the outcome of another.
• N — Number of trials is fixed: There is a predetermined, fixed count of trials (\(n\)).
• S — Success probability is constant: The probability of success (\(p\)) remains identical from trial to trial.
The Probability Mass Function (PMF)
To calculate the probability of getting exactly \(r\) successes out of \(n\) trials, we use the formula:
\(P(X = r) = \binom{n}{r} p^r (1 - p)^{n - r} = \binom{n}{r} p^r q^{n - r}\) for \(r = 0, 1, 2, \dots, n\)
Where the binomial coefficient is calculated as:
\(\binom{n}{r} = \frac{n!}{r!(n - r)!} = {}^n\text{C}_r\)
Why do we need \(\binom{n}{r}\)?
The term \(p^r (1 - p)^{n - r}\) only gives the probability of one specific order of outcomes (e.g., all successes first, then all failures). The combination factor \(\binom{n}{r}\) counts all the different possible orders in which those \(r\) successes could happen across the \(n\) trials!
Expectation (Mean) of a Binomial Distribution
The expected number of successes (the theoretical average or mean) over \(n\) trials is given by:
\(E(X) = \mu = np\)
Example: If you flip a fair coin 50 times (where \(p = 0.5\)), you expect to get \(E(X) = 50 \times 0.5 = 25\) heads.
Section 2 Key Takeaways:
• Model notation: \(X \sim B(n, p)\).
• Check the 4 conditions using BINS (Binary, Independent, Number fixed, Success constant).
• Single outcome formula: \(P(X = r) = \binom{n}{r} p^r (1 - p)^{n - r}\).
• Expected value: \(E(X) = np\).
3. Cumulative Probabilities & Inequality Translations
Translating Inequalities for Discrete Variables
Because discrete variables only take whole numbers (\(0, 1, 2, 3, \dots\)), the exact wording of a question matters tremendously. You must be able to convert phrases into cumulative notation (\(\le\)) to use your calculator efficiently.
Let's look at how to translate common phrases when \(X \sim B(n, p)\):
• "At most \(k\)" / "Up to \(k\)":
\(P(X \le k)\)
• "Fewer than \(k\)" / "Less than \(k\)":
\(P(X < k) = P(X \le k - 1)\)
• "At least \(k\)" / "No fewer than \(k\)":
\(P(X \ge k) = 1 - P(X \le k - 1)\)
• "More than \(k\)" / "Greater than \(k\)":
\(P(X > k) = 1 - P(X \le k)\)
• "Between \(a\) and \(b\) inclusive":
\(P(a \le X \le b) = P(X \le b) - P(X \le a - 1)\)
A Visual Number Line Trick
If you ever feel unsure, quickly write down the list of integers around your target number. Circle the ones you want and cross out the ones you do not want.
Example: If \(n = 10\) and you need \(P(X \ge 4)\):
Outcomes: \(0, 1, 2, 3 \mid 4, 5, 6, 7, 8, 9, 10\)
You want \(4\) through \(10\). The unwanted values are \(0, 1, 2, 3\), which is \(P(X \le 3)\).
Therefore: \(P(X \ge 4) = 1 - P(X \le 3)\).
Step-by-Step Worked Example
A fair 6-sided die is rolled 12 times. Let \(X\) be the number of times a '6' is rolled.
1. Identify the distribution:
\(X \sim B\left(12, \frac{1}{6}\right)\) where \(n = 12\) and \(p = \frac{1}{6} \approx 0.1667\).
2. Calculate the probability of getting exactly 2 sixes:
\(P(X = 2) = \binom{12}{2} \left(\frac{1}{6}\right)^2 \left(\frac{5}{6}\right)^{10}\)
\(P(X = 2) = 66 \times 0.02778 \times 0.16151 \approx 0.296\) (to 3 s.f.)
3. Calculate the probability of rolling fewer than 3 sixes:
\(P(X < 3) = P(X \le 2) = P(X = 0) + P(X = 1) + P(X = 2)\)
Using a calculator's cumulative binomial distribution function:
\(P(X \le 2) \approx 0.677\) (to 3 s.f.)
4. Calculate the probability of rolling at least 3 sixes:
\(P(X \ge 3) = 1 - P(X \le 2) = 1 - 0.6774 = 0.323\) (to 3 s.f.)
Section 3 Key Takeaways:
• For discrete variables, \(P(X < k)\) is not the same as \(P(X \le k)\).
• Convert all "at least" queries to \(1 - P(X \le k - 1)\).
• For range intervals, use \(P(a \le X \le b) = P(X \le b) - P(X \le a - 1)\).
4. Calculator Conventions & Common Pitfalls
Calculator Functions in CCEA AS 2
Under the CCEA 2210 specification, students are expected to use the built-in statistical distribution functions on modern scientific and graphical calculators rather than relying on paper tables:
• Binomial PD (Probability Density / Mass): Used to calculate exact probabilities for a single value, \(P(X = x)\).
• Binomial CD (Cumulative Distribution): Used to calculate cumulative probabilities, \(P(X \le x)\).
Top Examiner Warnings & Mistakes to Avoid
• Confusing the subtraction index: Writing \(P(X \ge 5) = 1 - P(X \le 5)\) instead of the correct form \(1 - P(X \le 4)\). Remember, you want to include 5, so you only subtract up to 4!
• Sampling Without Replacement: If items are drawn from a small finite group without replacement, the probability changes after each draw and trials are not independent. In this scenario, a Binomial model is not valid.
• Forgetting \(\binom{n}{r}\): Writing only \(p^r (1 - p)^{n - r}\) gives the probability of just one arrangement. You must include the combinatorial coefficient \(\binom{n}{r}\).
• Premature Rounding: Never round intermediate calculations. Keep the exact values in your calculator memory and round only your final answer to 3 significant figures unless specified otherwise.
Final Review Checklist:
✓ Have I checked all four BINS conditions before stating \(X \sim B(n, p)\)?
✓ Did I convert strict inequalities correctly before calculating cumulative probabilities?
✓ Is my final answer rounded accurately to 3 significant figures?