Let's go through each event and identify the outcomes that meet the criteria. Then, we will calculate the probability of each event.
Given outcomes:
[tex]\[
\begin{array}{c}
1. \; OOE \\
2. \; EEO \\
3. \; OEO \\
4. \; OEE \\
5. \; EEE \\
6. \; EOE \\
7. \; EOO \\
8. \; EOO \\
\end{array}
\][/tex]
Event A: An odd number on each of the first two rolls.
To meet this criteria, the first and second rolls must be odd. Let's check each outcome:
- OOE: First and second are odd (belongs to Event A)
- EEO: First is even
- OEO: Second is even
- OEE: Second is even
- EEE: First is even
- EOE: First is even
- EOO: First is even
- EOO: First is even
So, only OOE meets the criteria.
[tex]\[ \text{Number of favorable outcomes} = 1 \][/tex]
[tex]\[ \text{Total outcomes} = 8 \][/tex]
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{8} = 0.125 \][/tex]
Event B: Exactly one odd number.
To meet this criteria, the outcome must have exactly one 'O':
- OOE: Two odds
- EEO: One odd (belongs to Event B)
- OEO: Two odds
- OEE: One odd (belongs to Event B)
- EEE: No odds
- EOE: One odd (belongs to Event B)
- EOO: Two odds
- EOO: Two odds
So, EEO, OEE, and EOE meet the criteria.
[tex]\[ \text{Number of favorable outcomes} = 3 \][/tex]
[tex]\[ \text{Total outcomes} = 8 \][/tex]
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{3}{8} = 0.375 \][/tex]
Event C: Alternating even number and odd number (with either coming first).
For this criterion, the numbers must alternate and meet one of the following patterns: OEO or EOE:
- OOE: Not alternating
- EEO: Not alternating
- OEO: Alternating (belongs to Event C)
- OEE: Not alternating
- EEE: Not alternating
- EOE: Alternating (belongs to Event C)
- EOO: Not alternating
- EOO: Not alternating
So, OEO and EOE meet the criteria.
[tex]\[ \text{Number of favorable outcomes} = 2 \][/tex]
[tex]\[ \text{Total outcomes} = 8 \][/tex]
[tex]\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{2}{8} = 0.25 \][/tex]
Here is the completed table:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline & OOE & EEO & OEO & OEE & EEE & EOE & EOO & EOO & \text{Probability}\\
\hline \text{Event A: An odd number on each of} & \checkmark & & & & & & & & 0.125 \\
\hline \text{Event B: Exactly one odd number} & & \checkmark & & \checkmark & & \checkmark & & & 0.375 \\
\hline \text{Event C: Alternating even number and} & & & \checkmark & & & \checkmark & & & 0.25 \\
\hline
\end{array}
\][/tex]
This table correctly identifies the favorable outcomes for each event and their respective probabilities.
