Sample Space Diagram
What is a Sample Space?
The first step in any probability problem is to define the sample space, often denoted by the letter $S$. It is the "universe" of your experiment. An experiment is any process that leads to a well-defined result, like rolling a die, flipping a coin, or picking a card. An outcome is one single possible result from that experiment. For example, when flipping a coin, the sample space is $S = \{Heads, Tails\}$.
Outcomes are called equally likely if each has the same chance of occurring. A fair coin has equally likely outcomes. A sample space diagram is particularly powerful when we combine two or more simple experiments, creating a new, larger sample space.
Building Diagrams: Lists, Tables, and Trees
Sample space diagrams can take several forms, each suited to different types of experiments. The three most common are systematic lists, two-way tables, and tree diagrams.
1. Systematic Lists: For simple experiments, we can just list all outcomes. If you roll a standard six-sided die, the sample space is $S = \{1, 2, 3, 4, 5, 6\}$. For two dice, we list ordered pairs: $(1,1), (1,2), (1,3), ... (6,6)$.
2. Two-Way Tables: This is the classic "sample space diagram" for two-stage experiments. It organizes outcomes in a grid. Let's consider flipping two coins. The possible outcomes for the first coin are on one axis, and for the second coin on the other.
| Second Coin: H | Second Coin: T | |
|---|---|---|
| First Coin: H | (H, H) | (H, T) |
| First Coin: T | (T, H) | (T, T) |
The table clearly shows the 4 equally likely outcomes. We can now easily calculate probabilities. For example, the probability of getting one head and one tail (in any order) is $2/4 = 1/2$.
3. Tree Diagrams: For experiments that happen in stages, or have more than two parts, tree diagrams are excellent. Each branch represents a possible outcome at that stage. The endpoints of the tree (the "leaves") list the final outcomes.
Imagine a bag with a red (R) and a blue (B) marble. You pick one marble, note its color, put it back (this is called with replacement), and then pick again. The tree starts with two branches (R and B). From each of these, two more branches grow (R and B again). The four final outcomes are RR, RB, BR, BB.
From Simple Dice to Complex Decisions
Let's explore how sample space diagrams scale in complexity, using the universal example of dice.
Example 1: Rolling Two Dice
This is the most famous use of a two-way table. The sum of the two dice is a common event of interest. The sample space has $6 \times 6 = 36$ equally likely outcomes.
| + | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) sum=2 | (1,2) sum=3 | (1,3) sum=4 | (1,4) sum=5 | (1,5) sum=6 | (1,6) sum=7 |
| 2 | (2,1) sum=3 | (2,2) sum=4 | (2,3) sum=5 | (2,4) sum=6 | (2,5) sum=7 | (2,6) sum=8 |
From the full table (partially shown), we can count outcomes for any event. For example, the probability of rolling a sum of 7 is $6/36 = 1/6$, because the outcomes $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$ all yield a sum of 7.
Example 2: Dependent Events (Without Replacement)
Now, let's change the condition. What if you pick two marbles from a bag containing Red, Green, and Blue marbles without replacement (you don't put the first one back)? The second pick depends on the first. A tree diagram is perfect here.
Start: 3 branches: R, G, B.
If you picked R first, the second stage only has 2 branches: G, B.
If you picked G first, the second stage has branches: R, B, and so on.
The final sample space has $3 \times 2 = 6$ outcomes: RG, RB, GR, GB, BR, BG. These are still equally likely if each marble is identical and picked randomly.
Applying Diagrams to Solve Real Probability Puzzles
Sample space diagrams are not just academic exercises; they solve real puzzles and prevent common logical errors.
The Sibling Puzzle: A family has two children. What is the probability that both are boys? The naive sample space might seem to be {2 boys, 1 boy 1 girl, 2 girls}. But this is not correct for equally likely outcomes. The order matters (older/younger). The correct, equally likely sample space diagram (using B for boy, G for girl) is:
$S = \{ (B,B), (B,G), (G,B), (G,G) \}$.
Therefore, $P(\text{both boys}) = 1/4$.
Game Design: Imagine designing a simple board game where you flip a coin and roll a die to move. What is the chance of flipping "Heads" and rolling a number greater than 4? A two-way table with 2 coin outcomes and 6 die outcomes gives 12 total outcomes. The event "Heads and 5 or 6" corresponds to two specific cells in the table: (H,5) and (H,6). So, $P = 2/12 = 1/6$.
Important Questions
An outcome is one single, specific possible result of an experiment (e.g., rolling a 4). An event is a set of one or more outcomes that share a defined characteristic (e.g., rolling an even number, which includes the outcomes 2, 4, 6). An event is like a club, and outcomes are its members.
The formula $P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$ assumes each outcome has an equal "weight" or chance. If outcomes are not equally likely (like a weighted die), counting them gives a wrong answer. In such cases, we need to assign probabilities to individual outcomes first, often based on long-run frequency or given data.
Use a two-way table when your experiment has exactly two stages (like two dice, two coins, or picking one card from each of two decks) and the order matters. Use a tree diagram when you have more than two stages, or when the probabilities change at each stage (dependent events), as it clearly shows the branching possibilities and conditional probabilities on each branch.
The sample space diagram is more than just a list; it is a fundamental visualization tool that brings order to randomness. By forcing us to systematically account for every possible outcome, it eliminates guesswork and forms the solid ground upon which all basic probability calculations stand. From the simple flip of a coin to the strategic analysis of games, mastering the creation and use of these diagrams—whether as lists, tables, or trees—empowers students to think logically and solve problems with confidence. It transforms the abstract concept of "chance" into a concrete, countable, and manageable framework.
Footnote
1 Experiment: In probability, an experiment is any process or action that leads to a result that cannot be predicted with certainty beforehand (e.g., rolling a die, spinning a spinner).
2 Equally Likely: Describes outcomes that all have the exact same probability or chance of occurring. This is a key assumption in many introductory probability problems.
3 With Replacement: An experiment where a selected item is returned to the original set before the next selection, ensuring the sample space remains unchanged for subsequent actions.
4 Without Replacement: An experiment where a selected item is not returned. This changes the possible outcomes for subsequent actions, leading to dependent events.