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There are two groups of subjects, one of which consists of 5 science and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects, an unbiased die is cast. If the number 3 or 5 turns up a subject is selected at random from the first group. What is the probability that an engineering subject is ultimately selected?
- a)13/24
- b)5/12
- c)9/16
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Understanding the Problem
To find the probability of selecting an engineering subject after rolling an unbiased die, we need to consider the two groups of subjects and the outcomes of the die roll.
Die Outcomes
- The die has 6 faces with numbers 1 to 6.
- Outcomes that lead to selecting from Group 1: 3 or 5 (2 favorable outcomes).
- Probability of rolling a 3 or 5:
- P(3 or 5) = Number of favorable outcomes / Total outcomes = 2/6 = 1/3
Group 1 Composition
- Group 1 consists of 5 science and 3 engineering subjects.
- Total subjects in Group 1 = 5 + 3 = 8.
- Probability of selecting an engineering subject from Group 1:
- P(Engineering | Group 1) = Number of engineering subjects / Total subjects = 3/8
Calculating Overall Probability
To find the overall probability of selecting an engineering subject when the die shows 3 or 5 and we choose from Group 1, we multiply the probabilities:
- Overall Probability = P(3 or 5) * P(Engineering | Group 1)
- Overall Probability = (1/3) * (3/8) = 3/24 = 1/8
Conclusion
However, we also need to consider the probability of not rolling a 3 or 5, which would lead to selecting from Group 2. The outcomes leading to Group 2 are 1, 2, 4, or 6 (4 favorable outcomes).
Group 2 Composition
- Group 2 consists of 3 science and 5 engineering subjects.
- Total subjects in Group 2 = 3 + 5 = 8.
- Probability of selecting an engineering subject from Group 2:
- P(Engineering | Group 2) = 5/8
Final Calculation
- Probability of not rolling a 3 or 5: P(Not 3 or 5) = 4/6 = 2/3
- Overall Probability from Group 2 = (2/3) * (5/8) = 10/24
Total Probability of Selecting Engineering Subject
- Total Probability = Probability from Group 1 + Probability from Group 2
- Total Probability = (3/24) + (10/24) = 13/24
Thus, the correct answer is option 'A': 13/24.
To find the probability of selecting an engineering subject after rolling an unbiased die, we need to consider the two groups of subjects and the outcomes of the die roll.
Die Outcomes
- The die has 6 faces with numbers 1 to 6.
- Outcomes that lead to selecting from Group 1: 3 or 5 (2 favorable outcomes).
- Probability of rolling a 3 or 5:
- P(3 or 5) = Number of favorable outcomes / Total outcomes = 2/6 = 1/3
Group 1 Composition
- Group 1 consists of 5 science and 3 engineering subjects.
- Total subjects in Group 1 = 5 + 3 = 8.
- Probability of selecting an engineering subject from Group 1:
- P(Engineering | Group 1) = Number of engineering subjects / Total subjects = 3/8
Calculating Overall Probability
To find the overall probability of selecting an engineering subject when the die shows 3 or 5 and we choose from Group 1, we multiply the probabilities:
- Overall Probability = P(3 or 5) * P(Engineering | Group 1)
- Overall Probability = (1/3) * (3/8) = 3/24 = 1/8
Conclusion
However, we also need to consider the probability of not rolling a 3 or 5, which would lead to selecting from Group 2. The outcomes leading to Group 2 are 1, 2, 4, or 6 (4 favorable outcomes).
Group 2 Composition
- Group 2 consists of 3 science and 5 engineering subjects.
- Total subjects in Group 2 = 3 + 5 = 8.
- Probability of selecting an engineering subject from Group 2:
- P(Engineering | Group 2) = 5/8
Final Calculation
- Probability of not rolling a 3 or 5: P(Not 3 or 5) = 4/6 = 2/3
- Overall Probability from Group 2 = (2/3) * (5/8) = 10/24
Total Probability of Selecting Engineering Subject
- Total Probability = Probability from Group 1 + Probability from Group 2
- Total Probability = (3/24) + (10/24) = 13/24
Thus, the correct answer is option 'A': 13/24.
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There are two groups of subjects, one of which consists of 5 science and 3 engineering subjects and the other consists of 3 science and 5 engineering subjects, an unbiased die is cast. If the number 3 or 5 turns up a subject is selected at random from the first group. What is the probability that an engineering subject is ultimately selected?a)13/24b)5/12c)9/16d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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