Sure, let's work through the problem step by step. We will determine the outcomes for each event and their respective probabilities.
### Event A: An even number on both the first and the last rolls
Conditions:
- First roll: Even ('E')
- Last roll: Even ('E')
Possible Outcomes:
- EOE
- EEE
Probability Calculation:
There are 8 possible outcomes in total. The outcomes satisfying Event A are EOE and EEE.
- Number of favorable outcomes = 2
- Total number of possible outcomes = 8
Probability of Event A = (Number of favorable outcomes) / (Total number of possible outcomes)
[tex]\[ P(A) = \frac{2}{8} = 0.25 \][/tex]
### Event B: An odd number on each of the last two rolls
Conditions:
- Second roll: Odd ('O')
- Last roll: Odd ('O')
Possible Outcomes:
- OOO
- EOO
Probability Calculation:
There are 8 possible outcomes in total. The outcomes satisfying Event B are OOO and EOO.
- Number of favorable outcomes = 2
- Total number of possible outcomes = 8
Probability of Event B = (Number of favorable outcomes) / (Total number of possible outcomes)
[tex]\[ P(B) = \frac{2}{8} = 0.25 \][/tex]
### Event C: An even number on the last roll
Condition:
- Last roll: Even ('E')
Possible Outcomes:
- OOE
- OEE
- EOE
- EEE
Probability Calculation:
There are 8 possible outcomes in total. The outcomes satisfying Event C are OOE, OEE, EOE, and EEE.
- Number of favorable outcomes = 4
- Total number of possible outcomes = 8
Probability of Event C = (Number of favorable outcomes) / (Total number of possible outcomes)
[tex]\[ P(C) = \frac{4}{8} = 0.5 \][/tex]
### Summary of Results
| Event | Favorable Outcomes | Probability |
|----------------------------|---------------------------------|-------------|
| Event A: An even number on both the first and the last rolls | EOE, EEE | 0.25 |
| Event B: An odd number on each of the last two rolls | OOO, EOO | 0.25 |
| Event C: An even number on the last roll | OOE, OEE, EOE, EEE | 0.5 |
Thus, we have correctly identified the favorable outcomes and calculated the probabilities for each event.
