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
EOE EEO OEE OOE OOO EEE OEO EOO
Event A: Alternating even number and odd number (with either coming first)
Event B: An even number on both the first and the last rolls
Event C: No even numbers on the last two rolls

Answer 1

To analyze the events and their corresponding outcomes, we first identify the outcomes of rolling a number cube three times (where "E" denotes an even number and "O" denotes an odd number). The complete list of outcomes from three rolls is:

- EEE
- EEO
- EOE
- OEE
- OEO
- OOO
- OOE
- OEE

Now, let's check the outcomes that belong to each of the events A, B, and C.

### Event A: Alternating even number and odd number (with either coming first)
The outcomes that fit this event ((any arrangement of "E" and "O" such that they alternate)) are:

- EOE
- OEO

#### Outcomes for Event A:
- EOE
- OEO

#### Probability of Event A:
There are 2 outcomes out of the total of 8 possible outcomes. Therefore, the probability is:

\[
\frac{2}{8} = \frac{1}{4} \text{ or } 0.25
\]

---

### Event B: An even number on both the first and the last rolls
The outcomes that fit this event (even numbers on the first and last positions) are:

- EEO
- EOE
- OEE
- EEE

#### Outcomes for Event B:
- EEO
- EOE
- OEE
- EEE

#### Probability of Event B:
There are 4 outcomes out of 8. Therefore, the probability is:

\[
\frac{4}{8} = \frac{1}{2} \text{ or } 0.5
\]

---

### Event C: No even numbers on the last two rolls
The outcomes that fit this event (only odd numbers in the 2nd and 3rd positions) are:

- OOO
- OOE
- OEO

#### Outcomes for Event C:
- OOO
- OOE
- OEO

#### Probability of Event C:
There are 3 outcomes out of 8. Therefore, the probability is:

\[
\frac{3}{8} = 0.375
\]

---

Based on the analysis, here’s the final table:

| Events | Outcomes | Probability |
|-------------------------------------|------------------------------------------------------|-------------|
| Event A: Alternating odd/even | EOE, OEO | 0.25 |
| Event B: Even numbers first/last | EEO, EOE, OEE, EEE | 0.5 |
| Event C: No evens last two rolls | OOO, OOE, OEO | 0.375 |