Let's break down the problem into smaller steps to solve each part systematically:
### a. Number of elements in the sample space
Given that each die has faces with the numbers 9, 6, 3, and 2, we can calculate the sample space by considering all possible pairs of these values when two dice are thrown. Each die can land on any of its 4 faces, thus:
- Number of faces on die 1 × Number of faces on die 2 = 4 × 4 = 16.
Therefore, the sample space has 16 elements.
### b. Event "the total showing is even"
To find this event, we need to identify all pairs (a, b) such that when their numbers are added together, the sum is even.
Total showing is even when the sum of the numbers is divisible by 2. From the list of outcomes, the pairs meeting this criterion are:
- (9, 9)
- (9, 3)
- (6, 6)
- (6, 2)
- (3, 9)
- (3, 3)
- (2, 6)
- (2, 2)
### c. Probability that the total showing is even
The probability of an event occurring is given by the number of favorable outcomes divided by the total number of possible outcomes. From part (b), we've identified 8 pairs that result in an even total out of the 16 possible pairs.
So, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{8}{16} = 0.5. \][/tex]
### d. Probability that the total showing is greater than 11
Next, we find the pairs where the sum of the numbers is greater than 11:
- (9, 9)
- (9, 6)
- (9, 3)
- (6, 9)
- (6, 6)
- (3, 9)
There are 6 such pairs.
Thus, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{16} = 0.375. \][/tex]
### Problem Solving Method
The correct problem-solving method used here is:
D. The Be Systematic Principle
We systematically listed all possible outcomes and carefully selected the outcomes that match the criteria for each event.
By following these steps, we ensure that our calculations are comprehensive and accurate.
