Question
To win the game, Elena has to roll an even number first and a number less than 3 second. Her probability of winning is StartFraction 6 over 36 EndFraction.
A table with 36 total outcomes.
Marta has a lower probability of winning than Elena has. Which could be the outcome that Marta needs to win the game? Select three options.
rolling a sum of 7
rolling a sum of 6
rolling a sum of 2 or a sum of 9
rolling a sum that is greater than 9
rolling a sum that is greater than 2 but less than 5
A table with 36 total outcomes.
Marta has a lower probability of winning than Elena has. Which could be the outcome that Marta needs to win the game? Select three options.
rolling a sum of 7
rolling a sum of 6
rolling a sum of 2 or a sum of 9
rolling a sum that is greater than 9
rolling a sum that is greater than 2 but less than 5
Answers
GPT-4o mini
To help determine which outcomes Marta might need to win the game with a probability lower than Elena's \( \frac{6}{36} \) (or \( \frac{1}{6} \)), we can analyze each of the given options and their probabilities.
1. **Rolling a sum of 7**:
- The outcomes that yield a sum of 7 with two dice are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
- There are 6 ways to get a sum of 7.
- Probability = \( \frac{6}{36} = \frac{1}{6} \)
- This isn't lower than Elena's probability.
2. **Rolling a sum of 6**:
- The outcomes that yield a sum of 6 are: (1,5), (2,4), (3,3), (4,2), (5,1).
- There are 5 ways to get a sum of 6.
- Probability = \( \frac{5}{36} \)
- This is lower than \( \frac{6}{36} \).
3. **Rolling a sum of 2 or a sum of 9**:
- The outcomes for a sum of 2 is just (1,1).
- The outcomes for a sum of 9 are: (3,6), (4,5), (5,4), (6,3).
- Total outcomes: 1 (for sum of 2) + 4 (for sum of 9) = 5 outcomes total.
- Probability = \( \frac{5}{36} \)
- This is also lower than \( \frac{6}{36} \).
4. **Rolling a sum that is greater than 9**:
- The sums greater than 9 are 10, 11, and 12.
- The outcomes for those are:
- 10: (4,6), (5,5), (6,4) → 3 outcomes
- 11: (5,6), (6,5) → 2 outcomes
- 12: (6,6) → 1 outcome
- Total outcomes: 3 + 2 + 1 = 6 outcomes.
- Probability = \( \frac{6}{36} = \frac{1}{6} \)
- This isn’t lower than Elena's probability.
5. **Rolling a sum that is greater than 2 but less than 5**:
- The sums greater than 2 but less than 5 are 3 and 4.
- The outcomes are:
- 3: (1,2), (2,1) → 2 outcomes
- 4: (1,3), (2,2), (3,1) → 3 outcomes
- Total outcomes: 2 + 3 = 5 outcomes.
- Probability = \( \frac{5}{36} \)
- This is lower than \( \frac{6}{36} \).
Based on this analysis, the three options Marta could choose to win with a probability lower than Elena's are:
- Rolling a sum of 6
- Rolling a sum of 2 or a sum of 9
- Rolling a sum that is greater than 2 but less than 5
So, the correct answers are:
1. Rolling a sum of 6
2. Rolling a sum of 2 or a sum of 9
3. Rolling a sum that is greater than 2 but less than 5
1. **Rolling a sum of 7**:
- The outcomes that yield a sum of 7 with two dice are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
- There are 6 ways to get a sum of 7.
- Probability = \( \frac{6}{36} = \frac{1}{6} \)
- This isn't lower than Elena's probability.
2. **Rolling a sum of 6**:
- The outcomes that yield a sum of 6 are: (1,5), (2,4), (3,3), (4,2), (5,1).
- There are 5 ways to get a sum of 6.
- Probability = \( \frac{5}{36} \)
- This is lower than \( \frac{6}{36} \).
3. **Rolling a sum of 2 or a sum of 9**:
- The outcomes for a sum of 2 is just (1,1).
- The outcomes for a sum of 9 are: (3,6), (4,5), (5,4), (6,3).
- Total outcomes: 1 (for sum of 2) + 4 (for sum of 9) = 5 outcomes total.
- Probability = \( \frac{5}{36} \)
- This is also lower than \( \frac{6}{36} \).
4. **Rolling a sum that is greater than 9**:
- The sums greater than 9 are 10, 11, and 12.
- The outcomes for those are:
- 10: (4,6), (5,5), (6,4) → 3 outcomes
- 11: (5,6), (6,5) → 2 outcomes
- 12: (6,6) → 1 outcome
- Total outcomes: 3 + 2 + 1 = 6 outcomes.
- Probability = \( \frac{6}{36} = \frac{1}{6} \)
- This isn’t lower than Elena's probability.
5. **Rolling a sum that is greater than 2 but less than 5**:
- The sums greater than 2 but less than 5 are 3 and 4.
- The outcomes are:
- 3: (1,2), (2,1) → 2 outcomes
- 4: (1,3), (2,2), (3,1) → 3 outcomes
- Total outcomes: 2 + 3 = 5 outcomes.
- Probability = \( \frac{5}{36} \)
- This is lower than \( \frac{6}{36} \).
Based on this analysis, the three options Marta could choose to win with a probability lower than Elena's are:
- Rolling a sum of 6
- Rolling a sum of 2 or a sum of 9
- Rolling a sum that is greater than 2 but less than 5
So, the correct answers are:
1. Rolling a sum of 6
2. Rolling a sum of 2 or a sum of 9
3. Rolling a sum that is greater than 2 but less than 5