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Sagot :
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.
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.
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