To answer these questions, we need to follow a step-by-step process:
1. Use the Counting Principle:
The Counting Principle states that if there are m ways of doing one thing and n ways of doing another thing after it, then there are m x n ways of doing both things together.
In this case, there are 2 choices for the number (3 or 7), and 3 choices for each of the two letters (C, A, or R). Therefore, by using the counting principle, we can determine that there are:
2 (choices for the number) x 3 (choices for the first letter) x 3 (choices for the second letter) = 18 points in the sample space.
2. Construct a Tree Diagram:
To represent the situation, we can construct a tree diagram. Start by drawing two branches from the main point for the number (3 and 7). From each number branch, draw three branches for each letter choice (C, A, and R). The resulting tree diagram will have 18 endpoints representing all combinations of numbers and letters.
Number
/ \
3 7
/ \ / \
C A R C A R
3. List the Sample Space:
Based on the tree diagram, we can list all the possible combinations:
3CC, 3CA, 3CR, 3AC, 3AA, 3AR, 3RC, 3RA, 3RR
7CC, 7CA, 7CR, 7AC, 7AA, 7AR, 7RC, 7RA, 7RR
4. Determine the Exact Probability of Creating a Mini License Plate with a C:
We are looking for the probability of getting a C as the first letter in the mini license plate. We found that there are 18 possible combinations in the sample space.
Out of those 18 combinations, 6 start with the letter C (3CC, 3CA, 3CR, 7CC, 7CA, 7CR).
Therefore, the probability of creating a mini license plate with a C is 6/18. We can simplify this fraction to get the exact probability in reduced form:
6/18 = 1/3.
So, the exact probability of creating a mini license plate with a C is 1/3.