Alright, let's address the probabilities for the given scenarios when throwing a pair of unbiased dice. We will consider the detailed step-by-step solution for each case:
### Step-by-Step Solution
#### 1. Understanding the total number of outcomes:
When two unbiased dice are thrown, each die has 6 faces. Therefore, the total number of possible outcomes is:
[tex]\[ 6 \times 6 = 36 \][/tex]
#### 2. Probability that the dice show the same number:
For the dice to show the same number, both dice need to land on the same face. The possible pairs are:
[tex]\[ (1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \][/tex]
Thus, there are 6 favorable outcomes. The probability is given by:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{36} = 0.16666666666666666 \][/tex]
#### 3. Probability that the sum of the dice is 7:
Next, we find the pairs where the sum of the dice equals 7. These pairs are:
[tex]\[ (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) \][/tex]
There are 6 favorable outcomes for this event. Therefore, the probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{36} = 0.16666666666666666 \][/tex]
#### 4. Probability that the sum of the dice is at least 8:
Finally, we determine the pairs where the sum of the dice is 8 or more. These pairs are:
[tex]\[ \text{Sum 8:} \ (2,6), (3,5), (4,4), (5,3), (6,2) \][/tex]
[tex]\[ \text{Sum 9:} \ (3,6), (4,5), (5,4), (6,3) \][/tex]
[tex]\[ \text{Sum 10:} \ (4,6), (5,5), (6,4) \][/tex]
[tex]\[ \text{Sum 11:} \ (5,6), (6,5) \][/tex]
[tex]\[ \text{Sum 12:} \ (6,6) \][/tex]
Thus, there are 15 favorable outcomes. The probability is:
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{15}{36} = 0.4166666666666667 \][/tex]
### Summary of Probabilities:
- Probability that the dice show the same number: [tex]\( \frac{6}{36} = 0.16666666666666666 \)[/tex]
- Probability that the sum of the dice is 7: [tex]\( \frac{6}{36} = 0.16666666666666666 \)[/tex]
- Probability that the sum of the dice is at least 8: [tex]\( \frac{15}{36} = 0.4166666666666667 \)[/tex]
These are the detailed probabilities for the given scenarios when a pair of dice is thrown.
