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A pair of six-faced dice is rolled twice, The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts i s _____ (round off to three decimal places).
Correct answer is between '0.018,0.020'. Can you explain this answer?
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A pair of six-faced dice is rolled twice, The probability that the sum...
To find the probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts, we need to break down the problem into smaller steps.
Step 1: Calculate the total number of outcomes:
Each dice has six faces, so the total number of outcomes for a single roll is 6. Since we are rolling the pair of dice twice, the total number of outcomes for the two rolls is 6 * 6 = 36.
Step 2: Calculate the number of favorable outcomes:
In order to have a sum of 4 in a single roll, the possible combinations are:
- (1, 3)
- (2, 2)
- (3, 1)
Out of these three combinations, we need to find the number of combinations that occur exactly twice in three attempts. Let's analyze the possibilities:
Case 1: The combination occurs in the first two attempts:
The probability of this case happening is (1/36) * (1/36) = 1/1296.
Case 2: The combination occurs in the second and third attempts:
The probability of this case happening is (35/36) * (1/36) * (1/36) = 35/46656.
Case 3: The combination occurs in the first and third attempts:
The probability of this case happening is (1/36) * (35/36) * (1/36) = 35/46656.
Adding up the probabilities of all three cases, we get (1/1296) + (35/46656) + (35/46656) = 71/46656.
Step 3: Calculate the probability:
Since we are looking for the probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts, we need to multiply the probability of the favorable outcomes by the number of ways it can occur. In this case, there are three ways this can happen (one for each case).
Therefore, the probability is 3 * (71/46656) = 213/46656 ≈ 0.00457.
Final answer:
Rounding off to three decimal places, the probability is approximately 0.005.
Step 1: Calculate the total number of outcomes:
Each dice has six faces, so the total number of outcomes for a single roll is 6. Since we are rolling the pair of dice twice, the total number of outcomes for the two rolls is 6 * 6 = 36.
Step 2: Calculate the number of favorable outcomes:
In order to have a sum of 4 in a single roll, the possible combinations are:
- (1, 3)
- (2, 2)
- (3, 1)
Out of these three combinations, we need to find the number of combinations that occur exactly twice in three attempts. Let's analyze the possibilities:
Case 1: The combination occurs in the first two attempts:
The probability of this case happening is (1/36) * (1/36) = 1/1296.
Case 2: The combination occurs in the second and third attempts:
The probability of this case happening is (35/36) * (1/36) * (1/36) = 35/46656.
Case 3: The combination occurs in the first and third attempts:
The probability of this case happening is (1/36) * (35/36) * (1/36) = 35/46656.
Adding up the probabilities of all three cases, we get (1/1296) + (35/46656) + (35/46656) = 71/46656.
Step 3: Calculate the probability:
Since we are looking for the probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts, we need to multiply the probability of the favorable outcomes by the number of ways it can occur. In this case, there are three ways this can happen (one for each case).
Therefore, the probability is 3 * (71/46656) = 213/46656 ≈ 0.00457.
Final answer:
Rounding off to three decimal places, the probability is approximately 0.005.
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Community Answer
A pair of six-faced dice is rolled twice, The probability that the sum...
A = {( 1, 3), (3, 1), (2, 2)}
P(A) = 3/36 = 1/12
= 11/12
Required probability
P(A) = 3/36 = 1/12
= 11/12Required probability
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A pair of six-faced dice is rolled twice, The probability that the sum of the outcomes in each roll equals 4 in exactly two of the three attempts i s _____ (round off to three decimal places).Correct answer is between '0.018,0.020'. Can you explain this answer?
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