To solve this problem, we need to determine the outcomes that satisfy each event and calculate the probability for each event based on the outcomes listed.

Each outcome represents three rolls of a cube, labeled by whether each roll produced an odd (O) or even (E) number. There are 8 possible outcomes:

• EOO, OOE, OEE, EEO, OOO, EEE, OEO, and EOE.

Step-by-Step Solution

Event A: More even numbers than odd numbers

This event occurs if there are more E’s (even numbers) than O’s (odd numbers) in the outcome string. So, we are looking for outcomes with two or three E’s:

• Outcomes: OEE, EEO, EEE, EOE

Probability: There are 4 outcomes that satisfy Event A out of 8 total outcomes.
$\text{Probability} = \frac{4}{8} = 0.5$

Event B: An even number on the last roll or the second roll (or both)

For this event, an E must appear in the second or third position in the outcome string:

• Outcomes: EOO, OEE, EEO, EEE, OEO, EOE

Probability: There are 6 outcomes that satisfy Event B out of 8 total outcomes. $\text{Probability} = \frac{6}{8} = 0.75$

Event C: An even number on both the first and the last rolls

For this event, we need an E in both the first and third positions in the outcome string:

• Outcomes: EOE, EEE

Probability: There are 2 outcomes that satisfy Event C out of 8 total outcomes. $\text{Probability} = \frac{2}{8} = 0.25$

Summary

Now we can fill in the table based on this analysis:

Would you like further details on any part of this solution or the reasoning behind the probability calculations?

Additional Questions

1. What would the probability be if we rolled the cube four times instead?
2. How would the probabilities change if the outcomes were not equally likely?
3. Can you generalize the approach for any number of rolls with similar events?
4. What if each face of the cube had different probabilities for landing on even or odd numbers?
5. How do dependent events affect the calculation of probabilities in similar problems?

Tip

When calculating probabilities, always ensure the total number of outcomes is correct, as this is the denominator in your probability formula.