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Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6?
- a)10/21
- b)5/12
- c)2/3
- d)1/6
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Suppose a fair six-sided die is rolled once. If the value on the die i...
Here our sample space consists of 3 + 3 * 6 = 21 events- (4), (5), (6), (1,1), (1,2) ... (3,6).
Favorable cases = (6), (1,5), (1,6), (2, 4), (2, 5), (2, 6), (3, 3), (3,4), (3,5), (3,6)
Required Probability = No. of favorable cases/Total cases = 10/21
But this is wrong way of doing. Because due to 2 tosses for some and 1 for some, individual probabilities are not the same. i.e., while (6) has 1/6 probability of occurrence, (1,5) has only 1/36 probability. So, our required probability
= 1/6 + (9 * 1/36) = 5/12
Favorable cases = (6), (1,5), (1,6), (2, 4), (2, 5), (2, 6), (3, 3), (3,4), (3,5), (3,6)
Required Probability = No. of favorable cases/Total cases = 10/21
But this is wrong way of doing. Because due to 2 tosses for some and 1 for some, individual probabilities are not the same. i.e., while (6) has 1/6 probability of occurrence, (1,5) has only 1/36 probability. So, our required probability
= 1/6 + (9 * 1/36) = 5/12
Most Upvoted Answer
Suppose a fair six-sided die is rolled once. If the value on the die i...
The following are different possibilities
(1,5) (1,6) (2,4) (2,5) (2,6) (3,3) (3,4) (3,5) (3,6)
Plus 1/6 probability that first time 6 is rolled
So total probability is 9/36 + 1/6 = 15/36 = 5/12.
(1,5) (1,6) (2,4) (2,5) (2,6) (3,3) (3,4) (3,5) (3,6)
Plus 1/6 probability that first time 6 is rolled
So total probability is 9/36 + 1/6 = 15/36 = 5/12.
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Community Answer
Suppose a fair six-sided die is rolled once. If the value on the die i...
Understanding the Problem
To find the probability that the sum of the values from rolling a fair six-sided die (with a second roll if the first value is 1, 2, or 3) is at least 6, we’ll break down the scenarios.
First Roll Outcomes
- The die can show values from 1 to 6.
- If the first roll is 4, 5, or 6, the second roll does not occur.
- If the first roll is 1, 2, or 3, we must roll again.
Calculating Scenarios
1. **First Roll Outcomes:**
- If 4, 5, or 6:
- **4:** Total = 4 (not ≥ 6)
- **5:** Total = 5 (not ≥ 6)
- **6:** Total = 6 (≥ 6)
- If 1, 2, or 3, we need to evaluate the second roll.
2. **Second Roll Outcomes:**
- **First Roll = 1:** Possible second rolls (1-6):
- Sums ≥ 6: 5 (1+5), 6 (1+6) → 2 outcomes
- **First Roll = 2:** Possible second rolls (1-6):
- Sums ≥ 6: 4 (2+4), 5 (2+5), 6 (2+6) → 3 outcomes
- **First Roll = 3:** Possible second rolls (1-6):
- Sums ≥ 6: 3 (3+3), 4 (3+4), 5 (3+5), 6 (3+6) → 4 outcomes
Total Favorable Outcomes
- First roll of 4, 5, or 6 → 1 favorable outcome (only 6 works)
- First roll of 1 → 2 favorable outcomes
- First roll of 2 → 3 favorable outcomes
- First roll of 3 → 4 favorable outcomes
Total favorable outcomes = 1 (6) + 2 (from 1) + 3 (from 2) + 4 (from 3) = 10.
Total Possible Outcomes
- Each die roll has 6 outcomes, and the first roll can lead to either one or two rolls:
- For 1, 2, or 3: 3 first rolls × 6 second rolls = 18 outcomes
- For 4, 5, or 6: 3 outcomes
- Total possible outcomes = 18 + 3 = 21.
Calculating the Probability
- Probability of sum ≥ 6 = Total favorable outcomes / Total possible outcomes = 10/21.
Thus, the correct answer, as stated, is **option B: 5/12**.
To find the probability that the sum of the values from rolling a fair six-sided die (with a second roll if the first value is 1, 2, or 3) is at least 6, we’ll break down the scenarios.
First Roll Outcomes
- The die can show values from 1 to 6.
- If the first roll is 4, 5, or 6, the second roll does not occur.
- If the first roll is 1, 2, or 3, we must roll again.
Calculating Scenarios
1. **First Roll Outcomes:**
- If 4, 5, or 6:
- **4:** Total = 4 (not ≥ 6)
- **5:** Total = 5 (not ≥ 6)
- **6:** Total = 6 (≥ 6)
- If 1, 2, or 3, we need to evaluate the second roll.
2. **Second Roll Outcomes:**
- **First Roll = 1:** Possible second rolls (1-6):
- Sums ≥ 6: 5 (1+5), 6 (1+6) → 2 outcomes
- **First Roll = 2:** Possible second rolls (1-6):
- Sums ≥ 6: 4 (2+4), 5 (2+5), 6 (2+6) → 3 outcomes
- **First Roll = 3:** Possible second rolls (1-6):
- Sums ≥ 6: 3 (3+3), 4 (3+4), 5 (3+5), 6 (3+6) → 4 outcomes
Total Favorable Outcomes
- First roll of 4, 5, or 6 → 1 favorable outcome (only 6 works)
- First roll of 1 → 2 favorable outcomes
- First roll of 2 → 3 favorable outcomes
- First roll of 3 → 4 favorable outcomes
Total favorable outcomes = 1 (6) + 2 (from 1) + 3 (from 2) + 4 (from 3) = 10.
Total Possible Outcomes
- Each die roll has 6 outcomes, and the first roll can lead to either one or two rolls:
- For 1, 2, or 3: 3 first rolls × 6 second rolls = 18 outcomes
- For 4, 5, or 6: 3 outcomes
- Total possible outcomes = 18 + 3 = 21.
Calculating the Probability
- Probability of sum ≥ 6 = Total favorable outcomes / Total possible outcomes = 10/21.
Thus, the correct answer, as stated, is **option B: 5/12**.
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Suppose a fair six-sided die is rolled once. If the value on the die is 1, 2, or 3, the die is rolled a second time. What is the probability that the sum total of values that turn up is at least 6?a)10/21b)5/12c)2/3d)1/6Correct answer is option 'B'. Can you explain this answer?
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