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Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively?
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Box contains 5 white and 7 black balls. Two successive drawn of 3 ball...
Understanding the Problem
In a box, there are 5 white and 7 black balls, totaling 12 balls. We will calculate the probabilities for two scenarios: with replacement and without replacement.
1. Probability with Replacement
- In this case, after drawing 3 balls, they are placed back into the box before the next draw.
- Probability of drawing white balls in the first draw:
- Number of ways to choose 3 white balls = 5C3 = 10 (only if there are at least 3 white balls)
- Total ways to choose 3 balls from 12 = 12C3 = 220
- Probability = (5C3)/(12C3) = 10/220 = 1/22
- Probability of drawing black balls in the second draw:
- Number of ways to choose 3 black balls = 7C3 = 35
- Probability = (7C3)/(12C3) = 35/220 = 7/44
- Combined Probability:
- P(White in first & Black in second) = (1/22) * (7/44) = 7/968
2. Probability without Replacement
- Here, the balls drawn are not returned to the box.
- Probability of drawing white balls in the first draw:
- Total ways to choose 3 white balls = 5C3 = 10
- Probability = (5C3)/(12C3) = 10/220 = 1/22
- Probability of drawing black balls in the second draw:
- After drawing 3 white, there are now 2 white and 7 black left. Total = 9 balls.
- Number of ways to choose 3 black balls = 7C3 = 35
- Total ways to choose 3 balls from 9 = 9C3 = 84
- Probability = (7C3)/(9C3) = 35/84 = 5/12
- Combined Probability:
- P(White in first & Black in second) = (1/22) * (5/12) = 5/264
Conclusion
- With Replacement: 7/968
- Without Replacement: 5/264
These calculations provide a clear understanding of the probabilities involved in the two scenarios.
In a box, there are 5 white and 7 black balls, totaling 12 balls. We will calculate the probabilities for two scenarios: with replacement and without replacement.
1. Probability with Replacement
- In this case, after drawing 3 balls, they are placed back into the box before the next draw.
- Probability of drawing white balls in the first draw:
- Number of ways to choose 3 white balls = 5C3 = 10 (only if there are at least 3 white balls)
- Total ways to choose 3 balls from 12 = 12C3 = 220
- Probability = (5C3)/(12C3) = 10/220 = 1/22
- Probability of drawing black balls in the second draw:
- Number of ways to choose 3 black balls = 7C3 = 35
- Probability = (7C3)/(12C3) = 35/220 = 7/44
- Combined Probability:
- P(White in first & Black in second) = (1/22) * (7/44) = 7/968
2. Probability without Replacement
- Here, the balls drawn are not returned to the box.
- Probability of drawing white balls in the first draw:
- Total ways to choose 3 white balls = 5C3 = 10
- Probability = (5C3)/(12C3) = 10/220 = 1/22
- Probability of drawing black balls in the second draw:
- After drawing 3 white, there are now 2 white and 7 black left. Total = 9 balls.
- Number of ways to choose 3 black balls = 7C3 = 35
- Total ways to choose 3 balls from 9 = 9C3 = 84
- Probability = (7C3)/(9C3) = 35/84 = 5/12
- Combined Probability:
- P(White in first & Black in second) = (1/22) * (5/12) = 5/264
Conclusion
- With Replacement: 7/968
- Without Replacement: 5/264
These calculations provide a clear understanding of the probabilities involved in the two scenarios.
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Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively?
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Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively?.
Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Box contains 5 white and 7 black balls. Two successive drawn of 3 balls are made (i) with replacement (ii) without replacement. The probability that the first draw would produce white balls and the second draw would produce black balls are respectively?.
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