Discrete Probability Distributions
Introduction
Imagine students is playing a game where the result is not measured on a scale like height or weight, but counted in whole numbers, such as the number of heads in three coin tosses, the number of defective items in a box, or the number of goals scored in a match. These outcomes are examples of discrete random variables because they can be listed one by one. A discrete probability distribution tells us how likely each possible value is. 🎲
In this lesson, students will learn how to identify discrete probability distributions, describe them using key terminology, and calculate important values such as the mean and variance. This topic is a core part of Statistics and Probability in IB Mathematics: Analysis and Approaches HL because it helps model real-world situations where results come from counting rather than measuring. By the end, students should be able to interpret a distribution table, use formulas correctly, and connect the ideas to broader statistical reasoning.
What Makes a Distribution Discrete?
A random variable is a variable whose value depends on chance. If the random variable can only take separate, countable values, it is discrete. For example, if $X$ is the number of sixes in two rolls of a fair die, then the possible values are $0$, $1$, or $2$. That is discrete because there are only a few possible outcomes.
A discrete probability distribution assigns a probability to each possible value of $X$. For a valid distribution, two rules must always hold:
$$P(X=x) \ge 0$$
for every possible value $x$, and
$$\sum P(X=x)=1.$$
This means all probabilities must be non-negative and must add up to exactly $1$. If a table does not satisfy these conditions, it is not a valid probability distribution.
A very important idea is that the values of $X$ are outcomes of the random variable, while the probabilities describe how likely each outcome is. Do not confuse the values with their probabilities. For example, if $P(X=2)=0.3$, then $2$ is a value of the random variable and $0.3$ is the probability of getting that value.
Representing a Discrete Probability Distribution
Discrete probability distributions are often shown in a table. A typical table might look like this:
- values of the random variable $x$
- probabilities $P(X=x)$
For example, suppose $X$ is the number of rainy days in a three-day period in a certain place, with probabilities given by a model:
$$P(X=0)=0.2,\; P(X=1)=0.4,\; P(X=2)=0.3,\; P(X=3)=0.1$$
To check whether this is valid, add the probabilities:
$$0.2+0.4+0.3+0.1=1.0$$
Since the total is $1$, this is a valid discrete probability distribution.
Graphically, discrete distributions are often drawn with bar charts. Each bar stands for one possible value. This is different from continuous distributions, where graphs usually show areas under a curve. The gaps between values matter: in a discrete distribution, the random variable jumps from one value to the next. 📊
Expected Value, Mean, and Variance
One of the most important ideas in discrete probability is the expected value. The expected value is the long-run average value of the random variable. It is written as $E(X)$ or $\mu$ when referring to the mean of the distribution.
The formula is:
$$E(X)=\sum xP(X=x).$$
This means multiply each value of $x$ by its probability, then add the results.
Example 1: Finding the mean
Suppose a random variable $X$ has the following distribution:
$$X: 0,1,2$$
$$P(X=x): 0.5,0.3,0.2$$
Then
$$E(X)=0(0.5)+1(0.3)+2(0.2)=0+0.3+0.4=0.7.$$
So the mean of the distribution is $0.7$.
This does not mean every trial gives $0.7$. It means that over many repetitions, the average outcome would be close to $0.7$. This is a key distinction in probability.
The variance measures spread around the mean. A distribution with a large variance is more spread out, while a distribution with a small variance is more concentrated near the mean.
A commonly used formula is:
$$\operatorname{Var}(X)=E(X^2)-[E(X)]^2.$$
To find $E(X^2)$, compute
$$E(X^2)=\sum x^2P(X=x).$$
Then the standard deviation is
$$\sigma=\sqrt{\operatorname{Var}(X)}.$$
Example 2: Finding the variance
Using the same distribution:
$$X: 0,1,2$$
$$P(X=x): 0.5,0.3,0.2$$
First calculate
$$E(X^2)=0^2(0.5)+1^2(0.3)+2^2(0.2)=0+0.3+0.8=1.1.$$
Then
$$\operatorname{Var}(X)=1.1-(0.7)^2=1.1-0.49=0.61.$$
So the standard deviation is
$$\sigma=\sqrt{0.61}.$$
These calculations help students describe not just what outcomes are possible, but how those outcomes are spread out.
Common Discrete Distributions in IB
Two important discrete distributions appear often in IB Mathematics: Analysis and Approaches HL.
Binomial distribution
The binomial distribution is used when there are a fixed number of independent trials, each trial has only two outcomes, and the probability of success stays the same each time. The random variable $X$ counts the number of successes.
If $X\sim B(n,p)$, then $n$ is the number of trials and $p$ is the probability of success on each trial.
The probability formula is:
$$P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}.$$
This is very useful in exam-style problems. For example, if a student answers $5$ multiple-choice questions and each question has a $0.25$ chance of being answered correctly by guessing, then the number of correct answers can be modeled by a binomial distribution.
Poisson distribution
The Poisson distribution is used for counting the number of times an event happens in a fixed interval of time, area, or volume when events occur independently at a constant average rate. If $X\sim \operatorname{Poisson}(\lambda)$, then $\lambda$ is the mean number of events in the interval.
The formula is:
$$P(X=r)=\frac{e^{-\lambda}\lambda^r}{r!}.$$
An example is the number of customers entering a shop in $10$ minutes, or the number of messages received in an hour.
Although binomial and Poisson distributions are both discrete, they model different kinds of situations. The binomial distribution is about a fixed number of trials, while the Poisson distribution is about events happening in a continuous interval but counted as whole numbers.
Why Discrete Distributions Matter
Discrete probability distributions are used to make predictions, compare risks, and make decisions based on data. For example, a factory may use a discrete distribution to estimate how many defective items are likely in a shipment. A sports analyst may use one to predict the number of goals scored in a match. A school may use one to model how many students pass a quiz when guessing on multiple-choice questions. 🏭⚽📚
These models connect directly to the wider Statistics and Probability topic because they combine data reasoning with probability. In Statistics, students might study real data and summarize it using mean, variance, and graphs. In Probability, students uses theoretical models to describe randomness. Discrete probability distributions sit right between these two ideas because they provide a mathematical model for count-based data.
A strong IB answer often includes interpretation, not just calculation. For example, if the mean number of defective items is $0.8$, students should explain that this means the long-run average per box is less than one defective item, not that every box contains exactly $0.8$ defects.
Conclusion
Discrete probability distributions describe the probabilities of countable outcomes. They are built from random variables that take separate values, and they must satisfy $P(X=x)\ge 0$ and $\sum P(X=x)=1$. students should know how to find the mean using $E(X)=\sum xP(X=x)$, how to calculate variance using $\operatorname{Var}(X)=E(X^2)-[E(X)]^2$, and how to recognize major models such as the binomial and Poisson distributions. These ideas are important in IB Mathematics: Analysis and Approaches HL because they support reasoning about uncertainty in real-life situations and strengthen the connection between theory and data. ✅
Study Notes
- A discrete random variable takes countable values such as $0,1,2,\dots$.
- A discrete probability distribution gives each possible value $x$ a probability $P(X=x)$.
- Valid distributions satisfy $P(X=x)\ge 0$ and $\sum P(X=x)=1$.
- The mean or expected value is $E(X)=\sum xP(X=x)$.
- The variance is $\operatorname{Var}(X)=E(X^2)-[E(X)]^2$.
- The standard deviation is $\sigma=\sqrt{\operatorname{Var}(X)}$.
- The binomial distribution models a fixed number of independent success/failure trials.
- The Poisson distribution models counts of events in a fixed interval with average rate $\lambda$.
- Discrete distributions are important for real-world counting problems in science, business, sport, and everyday decision-making.