Instructions
Welcome, problem solver! Today, we're going to explore all the possible results, or "outcomes," of a situation using a special tool called an outcome table (sometimes called a sample space grid). This table helps you organize and see every single possibility, which makes calculating probabilities much easier.
An outcome table is a grid where the options for one event are listed along the top row, and the options for a second event are listed down the first column. By filling in the grid, you map out every possible combination.
Example: Flipping a Coin and Rolling a Die
Let's say you flip a standard coin and roll a standard 6-sided die. What are all the possible outcomes?
Step 1: Set up the table. Put the coin outcomes (Heads, Tails) along the side and the die outcomes (1, 2, 3, 4, 5, 6) along the top.
Step 2: Fill in the cells with the combined outcomes.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| Heads (H) | H, 1 | H, 2 | H, 3 | H, 4 | H, 5 | H, 6 |
| Tails (T) | T, 1 | T, 2 | T, 3 | T, 4 | T, 5 | T, 6 |
By looking at the table, you can quickly see there are 12 total possible outcomes. From here, you can find probabilities. For example, the probability of getting Heads and an even number (H,2; H,4; H,6) is 3 out of 12, or 3/12, which simplifies to 1/4.
Your Turn!
Problem 1: Lunch Combo
A cafe offers a lunch special where you choose one sandwich and one drink. The options are:
- Sandwiches: Ham, Turkey, Veggie
- Drinks: Water, Juice, Soda
A. Complete the outcome table below to show all possible lunch combinations.
| Water | Juice | Soda | |
|---|---|---|---|
| Ham | |||
| Turkey | |||
| Veggie |
B. Using your completed table, answer the following questions (express probabilities as a fraction):
- How many total lunch combinations are possible?
- What is the probability of choosing a Turkey sandwich and Juice?
- What is the probability of choosing any sandwich with Soda?
Problem 2: Double Spinner Game
You spin two spinners for a game. Spinner A is divided into 3 equal sections (1, 2, 3). Spinner B is divided into 4 equal sections (Red, Blue, Yellow, Green).
A. Create an outcome table in the space below to show all possible results of spinning both spinners once.
B. Using your table, find the probability of the following events:
- Spinning a 1 and the color Blue.
- Spinning an odd number (1 or 3) and any color.
- Spinning a number less than 3 and a primary color (Red, Blue, or Yellow).
Problem 3: The Sum of Two 4-Sided Dice
Imagine you have two 4-sided dice (tetrahedrons), with faces numbered 1, 2, 3, and 4. When you roll them, you are interested in the sum of the numbers that land face down.
A. Complete the outcome table below. The numbers in the main cells should be the sum of the outcomes from Die 1 and Die 2.
| + | 1 (Die 2) | 2 (Die 2) | 3 (Die 2) | 4 (Die 2) |
|---|---|---|---|---|
| 1 (Die 1) | ||||
| 2 (Die 1) | ||||
| 3 (Die 1) | ||||
| 4 (Die 1) |
B. Use your table of sums to find the following probabilities:
- What is the probability that the sum of the two dice is exactly 5?
- What is the probability that the sum is an odd number?
- What is the most likely sum to occur? What is its probability?
Answer Key
Problem 1: Lunch Combo
A. Completed Table:
| Water | Juice | Soda | |
|---|---|---|---|
| Ham | Ham, Water | Ham, Juice | Ham, Soda |
| Turkey | Turkey, Water | Turkey, Juice | Turkey, Soda |
| Veggie | Veggie, Water | Veggie, Juice | Veggie, Soda |
B. Answers:
- There are 9 total lunch combinations.
- The probability of choosing a Turkey sandwich and Juice is 1/9.
- There are 3 outcomes that include Soda (Ham, Turkey, Veggie). The probability is 3/9, which simplifies to 1/3.
Problem 2: Double Spinner Game
A. Completed Table:
| Red | Blue | Yellow | Green | |
|---|---|---|---|---|
| 1 | 1, Red | 1, Blue | 1, Yellow | 1, Green |
| 2 | 2, Red | 2, Blue | 2, Yellow | 2, Green |
| 3 | 3, Red | 3, Blue | 3, Yellow | 3, Green |
B. Answers: (Total outcomes = 12)
- The probability of spinning a 1 and Blue is 1/12.
- The odd numbers are 1 and 3. There are 4 outcomes for number 1 and 4 outcomes for number 3, making 8 favorable outcomes. The probability is 8/12, which simplifies to 2/3.
- Numbers less than 3 are 1 and 2. Primary colors are Red, Blue, Yellow. The favorable outcomes are (1,R), (1,B), (1,Y), (2,R), (2,B), (2,Y). That's 6 favorable outcomes. The probability is 6/12, which simplifies to 1/2.
Problem 3: The Sum of Two 4-Sided Dice
A. Completed Table:
| + | 1 (Die 2) | 2 (Die 2) | 3 (Die 2) | 4 (Die 2) |
|---|---|---|---|---|
| 1 (Die 1) | 2 | 3 | 4 | 5 |
| 2 (Die 1) | 3 | 4 | 5 | 6 |
| 3 (Die 1) | 4 | 5 | 6 | 7 |
| 4 (Die 1) | 5 | 6 | 7 | 8 |
B. Answers: (Total outcomes = 16)
- The sum 5 appears 4 times in the table. The probability is 4/16, which simplifies to 1/4.
- The odd sums are 3, 5, and 7. There are two 3s, four 5s, and two 7s. That is 2 + 4 + 2 = 8 odd sums. The probability is 8/16, which simplifies to 1/2.
- The most likely sum is 5, as it appears most often (4 times). Its probability is 4/16 or 1/4.