Problem 9 - Algebra 2

Question

Suppose you roll two number cubes. Graph the probability distribution for each sample space. {both numbers even, both numbers odd, one number even and the other odd}

Step-by-Step Solution

Answer

The probability for both numbers being even is 0.25, for both numbers being odd is 0.25, and for one number being even and the other odd is 0.5.

Step 1: Determine The Total Number Of Outcomes

Since a cube has 6 faces, the total possible outcomes when rolling two number cubes is 6^2 = 36.

Step 2: Calculate The Possible Outcomes For Each Sample Space

Calculating each sample space: For both numbers even, there are 3 even numbers on a cube. Therefore, the number of favourable outcomes is 3*3 = 9. For both numbers odd, there are 3 odd numbers on a cube. Therefore, the number of favourable outcomes is 3*3 = 9. For one number even and one number odd, there are 3 even numbers and 3 odd numbers. Therefore, the number of favourable outcomes is 3*3 = 9 * 2 (as either of the cube can be even) = 18.

Step 3: Calculate The Probability For Each Sample Space

Probability is calculated by dividing the number of favourable outcomes by the total number of outcomes. For both numbers even, the probability is 9/36 = 0.25. For both numbers odd, the probability is 9/36 = 0.25. For one number even and one number odd, the probability is 18/36 = 0.5.

Step 4: Draw The Probability Distribution Graph

On the x-axis, list each sample space (both numbers even, both numbers odd, one number even and the other odd). On the y-axis, list the probability. Draw a bar above each sample space representing the calculated probability.

Key Concepts

Sample Space

In probability, the term 'sample space' refers to the set of all possible outcomes for a given experiment. For the exercise with two number cubes, the sample space is all potential outcomes when both cubes are rolled.
This includes every pair of numbers that each die can show, resulting in a total of 36 possible outcomes because each die has 6 sides.

• Both numbers even: Examples are (2,2), (2,4), etc. There are 9 such combinations.
• Both numbers odd: Examples are (1,1), (1,3), etc. Again, there are 9 combinations.
• One number even, other odd: Examples include (1,2), (2,1), yielding 18 such outcomes.

Understanding sample spaces is crucial as it sets the stage for performing probability calculations by providing the foundation of all potential outcomes in a scenario.

Combinatorics

Combinatorics is a branch of mathematics concerned with counting, arrangement, and combination of objects. In this exercise, combinatorics helps determine the number of ways outcomes can occur.
When rolling two dice, combinatorics allows us to confirm that there are 9 combinations for both even numbers and both odd numbers.

Counting Combinations

• For even numbers, if each die has 3 possible even numbers (2, 4, 6), the number of combinations is calculated as 3 even on one die times 3 even on the other die, totaling 9.
• Similarly, for odd numbers (1, 3, 5), the calculation is the same, resulting in 9 combinations.
• For mixed even and odd outcomes, either die can show 3 odd and 3 even numbers, so the combinations are 3 odd times 3 even, doubled because either die can be even, resulting in 18.

Combinatorics thus provides the essential counting capability needed for probability calculations.

Probability Calculation

Probability calculation involves determining how likely an event is to happen. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

In our exercise, since there are a total of 36 outcomes when rolling two dice (6 sides on each die), we can express probabilities as:

• For both numbers even: The probability is 9 (favorable outcomes) divided by 36 (total outcomes) = 0.25.
• For both numbers odd: Similarly, 9/36 = 0.25.
• For one even, one odd: 18 favorable outcomes out of 36 possibilities gives a probability of 0.5.

Graphing these values involves representing each scenario's probability as a bar, showing visually how often each event can be expected to occur. Understanding probability calculations allows us to quantify the likelihood of different outcomes based on a sample space.