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 8 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

Answer 1

To analyze the three events described, we will first clarify the meaning of the outcomes and probabilities.

We have a number cube (die) with 6 faces, where 3 faces show odd numbers (1, 3, 5) and 3 faces show even numbers (2, 4, 6). Each outcome in the list corresponds to a sequence of results from rolling the die three times, with 'O' representing an odd number and 'E' representing an even number. The possible outcomes are:
- OOE
- EEE
- OEO
- EOE
- OEE
- EEO
- OOO
- EOO

Now, let's identify the outcomes contained in each event and calculate the probabilities.

### Event A: An even number on the last roll or the second roll (or both)
**Outcomes that meet Event A:**
- **OOE** (last roll is E)
- **EEE** (last roll and second roll are both E)
- **OEO** (last roll is O, second roll is E)
- **EOE** (last roll is E)
- **OEE** (last roll is E)
- **EEO** (last roll is O, second roll is E)
- **EOO** (last roll is O, second roll is E)

Thus, the outcomes for Event A are: OOE, EEE, OEO, EOE, OEE, EEO, EOO.
In total, **7 outcomes** meet the criteria of Event A.

### Probability of Event A:
\[
P(A) = \frac{7}{8}
\]

### Event B: Two or more odd numbers
**Outcomes that meet Event B:**
- **OOE** (2 odd)
- **OOO** (3 odd)
- **EOO** (2 odd)

Thus, the outcomes for Event B are: OOE, OOO, EOO.
In total, **3 outcomes** meet the criteria of Event B.

### Probability of Event B:
\[
P(B) = \frac{3}{8}
\]

### Event C: An even number on both the first and the last rolls
**Outcomes that meet Event C:**
- **EEO** (1st roll is E, last roll is E)
- **EOE** (1st roll is E, last roll is E)
- **EEE** (1st roll is E, last roll is E)

Thus, the outcomes for Event C are: EEO, EOE, EEE.
In total, **3 outcomes** meet the criteria of Event C.

### Probability of Event C:
\[
P(C) = \frac{3}{8}
\]

### Summary Table

| Events | Outcomes Count | Probability |
|---------------------------------------------|----------------|-------------|
| Event A: An even number on the last roll or the second roll (or both) | 7 | 7/8 |
| Event B: Two or more odd numbers | 3 | 3/8 |
| Event C: An even number on both the first and the last rolls | 3 | 3/8 |

Thus, we have the final results for all events and their probabilities.