Compound Events, Tree Diagrams, Tables, and Organized Lists - Seventh Grade Mathematics
Compound events combine multiple independent or sequential actions, analyzed by constructing complete sample spaces using organized lists, coordinate grids, and branching tree diagrams.
Defining Compound Events
A simple event consists of a single outcome from a single random action, such as rolling a single die. A compound event consists of two or more simple events occurring in combination or succession, such as flipping a coin AND rolling a die, or drawing two cards in sequence.
SIMPLE vs COMPOUND EVENTS
Simple Event: Flipping heads on one coin: { H }
Compound Event: Flipping heads on coin AND { (H, 4) }
rolling a 4 on a die
Finding the probability of a compound event requires determining the complete sample space of all possible combined outcomes.
Systematic Representation Methods
To ensure no compound outcomes are accidentally overlooked or duplicated, mathematicians employ three systematic organizational tools:
Method 1: Organized Lists
An organized list arranges outcomes in a strict alphabetical or numerical sequence, systematically pairing each possibility of the first event with every possibility of subsequent events.
Experiment: Flipping a coin (H, T) and choosing a shirt color (Red, Blue, Green)
Organized List:
1. (H, Red) 4. (T, Red)
2. (H, Blue) 5. (T, Blue)
3. (H, Green) 6. (T, Green)
Total Possible Outcomes: 6
Method 2: Two-Way Tables and Grids
A two-way table is ideal when combining exactly two events, placing the outcomes of the first event along the rows and the outcomes of the second event along the columns.
Experiment: Rolling two standard six-sided dice (Sum Table)
Die 2: 1 2 3 4 5 6
Die 1: +-----+-----+-----+-----+-----+-----+
1 | 2 | 3 | 4 | 5 | 6 | 7 |
2 | 3 | 4 | 5 | 6 | 7 | 8 |
3 | 4 | 5 | 6 | 7 | 8 | 9 |
4 | 5 | 6 | 7 | 8 | 9 | 10 |
5 | 6 | 7 | 8 | 9 | 10 | 11 |
6 | 7 | 8 | 9 | 10 | 11 | 12 |
+-----+-----+-----+-----+-----+-----+
Total Outcomes = 6 rows * 6 columns = 36 total outcomes.
Using this grid, we see that a sum of seven appears six times along the diagonal. Thus, P(Sum of 7) = 6 / 36 = 1 / 6.
Method 3: Tree Diagrams
A tree diagram illustrates multi-stage events through branching paths. Each stage of the experiment forms a new layer of branches.
Experiment: Flipping a coin twice
Stage 1 (First Flip) Stage 2 (Second Flip) Final Outcome
/--- H -------------> (H, H)
/--- H (Heads) -------+
/ \--- T -------------> (H, T)
Start -+
\ /--- H -------------> (T, H)
\--- T (Tails) -------+
\--- T -------------> (T, T)
Total Paths from Start to Leaf = 4 distinct outcomes.
Following each path from the starting root to a final leaf identifies one unique compound outcome.
The Fundamental Counting Principle
The Fundamental Counting Principle provides a rapid mathematical shortcut for calculating the total size of a compound sample space without writing out every outcome.
THE FUNDAMENTAL COUNTING PRINCIPLE
If Event 1 can occur in m ways, and Event 2 can occur in n ways,
and Event 3 can occur in p ways:
Total Outcomes = m * n * p
Consider choosing a school outfit from four pairs of pants, five shirts, and three pairs of shoes: Total possible outfits: four times five times three equals sixty different outfits.
Exercises
Work through each problem carefully in your notebook before reading the provided solutions.
Exercise 1. A student flips a coin and rolls a standard six-sided die. Create an organized list of all possible compound outcomes, and find the probability of getting tails and an even number.
Exercise 2. A lunch combo offers a choice of sandwich (turkey, ham, veggie), a side (chips, apple), and a beverage (water, juice, milk). Use the Fundamental Counting Principle to determine the total number of distinct lunch combos available.
Exercise 3. Construct a tree diagram showing all possible outcomes when a fair coin is flipped three times. Use the diagram to calculate the probability of obtaining exactly two heads.
Exercise 4. Using the two-way sum table for rolling two standard six-sided dice, calculate the probability of rolling a sum strictly greater than nine.
Solutions
Solution to Exercise 1. Write the organized list systematically pairing each coin side (H, T) with die numbers 1 through 6: (H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6). There are twelve total outcomes (2 * 6 = 12). Identify the favorable outcomes matching tails and an even number: (T, 2), (T, 4), (T, 6). There are three favorable outcomes. Calculate the probability: three-twelfths, which simplifies to one-fourth (1/4 = 0.25 = 25%).
Solution to Exercise 2. Apply the Fundamental Counting Principle: multiply the number of options available at each stage. Sandwiches: 3 options. Sides: 2 options. Beverages: 3 options. Total combos: three times two times three. Three times two is six; six times three equals eighteen. There are eighteen distinct lunch combos available.
Solution to Exercise 3. The tree diagram starts with two branches for flip one: Heads (H) and Tails (T). Each branch splits into two for flip two: HH, HT, TH, TT. Each of those four branches splits into two for flip three, generating eight final paths: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. The total sample space contains eight outcomes. Identify the outcomes with exactly two heads: HHT, HTH, and THH. There are three favorable outcomes. The probability of obtaining exactly two heads is three-eighths (3/8 = 0.375 = 37.5%).
Solution to Exercise 4. The two-way grid for rolling two six-sided dice contains thirty-six total outcomes (6 * 6 = 36). Identify the cells where the sum is strictly greater than nine: sums of ten: (4, 6), (5, 5), (6, 4) (3 outcomes); sums of eleven: (5, 6), (6, 5) (2 outcomes); sums of twelve: (6, 6) (1 outcome). Sum the counts: three plus two plus one equals six favorable outcomes. Calculate the probability: six thirty-sixths, which simplifies to one-sixth (1/6 = 0.166... or approximately 16.7%).