Let's analyze the given events and determine the probabilities for each.
### Event A: An even number on both the first and the last rolls
We need to find outcomes that have even numbers in both the first and the last positions. Looking at all the possible outcomes, we see:
- EEE (even, even, even)
- EOE (even, odd, even)
- EEO (even, even, odd)
So the outcomes that satisfy Event A are EEE, EOE, and EEO. There are 3 such outcomes.
The probability of Event A is calculated as the number of favorable outcomes divided by the total number of outcomes:
[tex]\[ \text{Probability of Event A} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{3}{8} = 0.375 \][/tex]
### Event B: An even number on the second roll
We need to find outcomes that have an even number in the second position. Looking at all possible outcomes, we see:
- EEE (even, even, even)
- EOE (even, odd, even)
- EEO (even, even, odd)
- OEE (odd, even, even)
- OEO (odd, even, odd)
- OOE (odd, odd, even)
So the outcomes that satisfy Event B are EEE, EOE, EEO, OEE, OEO, and OOE. There are 6 such outcomes.
The probability of Event B is:
[tex]\[ \text{Probability of Event B} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{6}{8} = 0.75 \][/tex]
### Event C: No even numbers on the last two rolls
We need to find outcomes where the last two positions are occupied by odd numbers. Looking at all possible outcomes, we see:
- EOO (even, odd, odd)
- OOO (odd, odd, odd)
So the outcomes that satisfy Event C are EOO and OOO. There are 2 such outcomes.
The probability of Event C is:
[tex]\[ \text{Probability of Event C} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{8} = 0.25 \][/tex]
In summary, the probabilities are as follows:
- Event A: 0.375
- Event B: 0.75
- Event C: 0.25
