Question
A number cube is rolled three times. An outcome is represented by a string of the sort OEE (meaning an odd number on the first roll, an even number on the second roll, and an even number on the third roll). The outcomes are listed in the table below. Note that each outcome has the same probability.
For each of the three events in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.
Outcomes Probability
OOE EEE OEO EOE OEE EEO OOO EOO
Event A: An even number on the last roll or the second roll (or both)
Event B: Two or more odd numbers
Event C: An even number on both the first and the last rolls
For each of the three events in the table, check the outcome(s) that are contained in the event. Then, in the last column, enter the probability of the event.
Outcomes Probability
OOE EEE OEO EOE OEE EEO OOO EOO
Event A: An even number on the last roll or the second roll (or both)
Event B: Two or more odd numbers
Event C: An even number on both the first and the last rolls
Answers
GPT-4o mini
To solve this problem, we first be aware of the possible outcomes from rolling a number cube (die) three times, which are strings consisting of "O" for odd numbers (1, 3, 5) and "E" for even numbers (2, 4, 6). The outcomes of rolling a number cube three times consist of all possible combinations of "O" and "E".
Given the list of outcomes:
- OOE
- EEE
- OEO
- EOE
- OEE
- EEO
- OOO
- EOO
Now, let's analyze each event:
### Event A: An even number on the last roll or the second roll (or both)
Possible outcomes that meet this condition:
- OOE (last roll is E)
- EEE (last roll is E)
- OEO (second roll is E)
- EOE (second roll is E)
- OEE (last roll is E)
- EEO (second roll is E and last roll is E)
- OOO (no E but excludes)
- EOO (last roll is E)
The outcomes that meet Event A: **OOE, EEE, OEO, EOE, OEE, EEO, EOO**
### Event B: Two or more odd numbers
Possible outcomes that meet this condition:
- OOE (2 odd numbers)
- OEO (2 odd numbers)
- OEE (2 odd numbers)
- OOO (3 odd numbers)
- EOO (2 odd numbers)
The outcomes that meet Event B: **OOE, OEO, OEE, OOO, EOO**
### Event C: An even number on both the first and the last rolls
Possible outcomes that meet this condition:
- EEE (first and last are E)
- EEO (first and last are E)
- EOE (first and last are E)
The outcomes that meet Event C: **EEE, EEO, EOE**
### Probability Calculation
Considering that there are 8 total possible outcomes, the probability for each event can now be calculated based on the number of successful outcomes divided by the total number of outcomes (8).
- Probability of Event A: \( \frac{7}{8} \)
- Probability of Event B: \( \frac{5}{8} \)
- Probability of Event C: \( \frac{3}{8} \)
### Final Table
| Outcomes | Probability |
|----------|--------------|
| Event A | \( \frac{7}{8} \) |
| Event B | \( \frac{5}{8} \) |
| Event C | \( \frac{3}{8} \) |
This table summarizes the events and their probabilities based on our calculations.
Given the list of outcomes:
- OOE
- EEE
- OEO
- EOE
- OEE
- EEO
- OOO
- EOO
Now, let's analyze each event:
### Event A: An even number on the last roll or the second roll (or both)
Possible outcomes that meet this condition:
- OOE (last roll is E)
- EEE (last roll is E)
- OEO (second roll is E)
- EOE (second roll is E)
- OEE (last roll is E)
- EEO (second roll is E and last roll is E)
- OOO (no E but excludes)
- EOO (last roll is E)
The outcomes that meet Event A: **OOE, EEE, OEO, EOE, OEE, EEO, EOO**
### Event B: Two or more odd numbers
Possible outcomes that meet this condition:
- OOE (2 odd numbers)
- OEO (2 odd numbers)
- OEE (2 odd numbers)
- OOO (3 odd numbers)
- EOO (2 odd numbers)
The outcomes that meet Event B: **OOE, OEO, OEE, OOO, EOO**
### Event C: An even number on both the first and the last rolls
Possible outcomes that meet this condition:
- EEE (first and last are E)
- EEO (first and last are E)
- EOE (first and last are E)
The outcomes that meet Event C: **EEE, EEO, EOE**
### Probability Calculation
Considering that there are 8 total possible outcomes, the probability for each event can now be calculated based on the number of successful outcomes divided by the total number of outcomes (8).
- Probability of Event A: \( \frac{7}{8} \)
- Probability of Event B: \( \frac{5}{8} \)
- Probability of Event C: \( \frac{3}{8} \)
### Final Table
| Outcomes | Probability |
|----------|--------------|
| Event A | \( \frac{7}{8} \) |
| Event B | \( \frac{5}{8} \) |
| Event C | \( \frac{3}{8} \) |
This table summarizes the events and their probabilities based on our calculations.