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.

Question

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 OOE OEE OOO EOO EEE OEO
Event A: An even number on both the first and the last rolls
Event B: Two or more odd numbers
Event C: An even number on the second roll

Answer 1

To display the outcomes contained in each event and the associated probabilities, let's first break down the outcomes in relation to the events provided.

Outcomes

The possible outcomes generated by the rolls of the number cube are:

• EOE
• EEO
• OOE
• OEE
• OOO
• EOO
• EEE
• OEO

Analyzing Each Event

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

For an outcome to fulfill this condition, the first roll (1st character) and the last roll (3rd character) must both be even. From the provided outcomes:

• EOE (E - first, E - last)
• EEO (E - first, E - last)
• OOE (O - first, E - last)
• OEE (O - first, E - last)
• OOO (O - first, O - last)
• EOO (E - first, O - last)
• EEE (E - first, E - last)
• OEO (O - first, O - last)

Only EEO and EEE satisfy this condition, so:

• Outcomes: EEO, EEE
• Probability: 2 out of 8 = \( \frac{2}{8} = \frac{1}{4} \)

Event B: Two or more odd numbers

For an outcome to fulfill this condition, we need at least two odd numbers. The odd numbers are represented by 'O'. Analyzing the outcomes:

• EOE (1 odd)
• EEO (0 odd)
• OOE (2 odd)
• OEE (1 odd)
• OOO (3 odd)
• EOO (2 odd)
• EEE (0 odd)
• OEO (2 odd)

The outcomes that satisfy this condition are:

• OOE
• OOO
• EOO
• OEO

So:

• Outcomes: OOE, OOO, EOO, OEO
• Probability: 4 out of 8 = \( \frac{4}{8} = \frac{1}{2} \)

Event C: An even number on the second roll

For an outcome to fulfill this condition, the second roll (2nd character) must be even (E). Analyzing the outcomes:

• EOE (E)
• EEO (E)
• OOE (E)
• OEE (E)
• OOO (O)
• EOO (E)
• EEE (E)
• OEO (E)

The outcomes that satisfy this condition are:

• EOE
• EEO
• OOE
• OEE
• EOO
• EEE
• OEO

So:

• Outcomes: EOE, EEO, OOE, OEE, EOO, EEE, OEO
• Probability: 7 out of 8 = \( \frac{7}{8} \)

Summary Table

| Outcomes | Probability | |------------------|-------------| | Event A: EEO, EEE | \( \frac{1}{4} \) | | Event B: OOE, OOO, EOO, OEO | \( \frac{1}{2} \) | | Event C: EOE, EEO, OOE, OEE, EOO, EEE, OEO | \( \frac{7}{8} \) |

This summarizes the outcomes contained within each event and their respective probabilities.