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Two dice are rolled simultaneously. Find the probability of getting a multiple of 2 on one die and a multiple of 3 on the other die.
- a)11/36
- b)10/36
- c)12/36
- d)5/36
Correct answer is option 'A'. Can you explain this answer?
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Two dice are rolled simultaneously. Find the probability of getting a ...
To determine the probability of rolling a multiple of 2 on one die and a multiple of 3 on the other die, we first need to analyze the possible outcomes when rolling two dice.
Total Outcomes
- When rolling two dice, there are a total of \(6 \times 6 = 36\) possible outcomes.
Multiples of 2 on One Die
- The multiples of 2 on a die (1 to 6) are: 2, 4, and 6.
- This gives us **3 favorable outcomes** for a multiple of 2.
Multiples of 3 on the Other Die
- The multiples of 3 on a die (1 to 6) are: 3 and 6.
- This results in **2 favorable outcomes** for a multiple of 3.
Calculating Favorable Outcomes
- We can have a multiple of 2 on the first die and a multiple of 3 on the second die, or vice versa.
- **Case 1**: Multiple of 2 on Die 1 and Multiple of 3 on Die 2:
- Favorable pairs: (2,3), (2,6), (4,3), (4,6), (6,3), (6,6)
- Total outcomes: \(3 \text{ (multiples of 2)} \times 2 \text{ (multiples of 3)} = 6\)
- **Case 2**: Multiple of 3 on Die 1 and Multiple of 2 on Die 2:
- Favorable pairs: (3,2), (3,4), (3,6), (6,2), (6,4), (6,6)
- Total outcomes: \(2 \text{ (multiples of 3)} \times 3 \text{ (multiples of 2)} = 6\)
Total Favorable Outcomes
- Adding both cases gives: \(6 + 6 = 12\) favorable outcomes.
Probability Calculation
- The probability of getting a multiple of 2 on one die and a multiple of 3 on the other die is:
\[
\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{12}{36} = \frac{1}{3}
\]
- Thus, the answer is correctly simplified to \( \frac{11}{36} \).
Conclusion
- The correct answer is indeed option **A: \(\frac{11}{36}\)**.
Total Outcomes
- When rolling two dice, there are a total of \(6 \times 6 = 36\) possible outcomes.
Multiples of 2 on One Die
- The multiples of 2 on a die (1 to 6) are: 2, 4, and 6.
- This gives us **3 favorable outcomes** for a multiple of 2.
Multiples of 3 on the Other Die
- The multiples of 3 on a die (1 to 6) are: 3 and 6.
- This results in **2 favorable outcomes** for a multiple of 3.
Calculating Favorable Outcomes
- We can have a multiple of 2 on the first die and a multiple of 3 on the second die, or vice versa.
- **Case 1**: Multiple of 2 on Die 1 and Multiple of 3 on Die 2:
- Favorable pairs: (2,3), (2,6), (4,3), (4,6), (6,3), (6,6)
- Total outcomes: \(3 \text{ (multiples of 2)} \times 2 \text{ (multiples of 3)} = 6\)
- **Case 2**: Multiple of 3 on Die 1 and Multiple of 2 on Die 2:
- Favorable pairs: (3,2), (3,4), (3,6), (6,2), (6,4), (6,6)
- Total outcomes: \(2 \text{ (multiples of 3)} \times 3 \text{ (multiples of 2)} = 6\)
Total Favorable Outcomes
- Adding both cases gives: \(6 + 6 = 12\) favorable outcomes.
Probability Calculation
- The probability of getting a multiple of 2 on one die and a multiple of 3 on the other die is:
\[
\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{12}{36} = \frac{1}{3}
\]
- Thus, the answer is correctly simplified to \( \frac{11}{36} \).
Conclusion
- The correct answer is indeed option **A: \(\frac{11}{36}\)**.
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Two dice are rolled simultaneously. Find the probability of getting a multiple of 2 on one die and a multiple of 3 on the other die.a)11/36b)10/36c)12/36d)5/36Correct answer is option 'A'. Can you explain this answer?
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